1

Arthur E. Fitzgerald, Charles Kingsley, Jr., and Stephen D. Umans, Electric Machinery, Sixth Edition, McGraw-Hill, 2003.

1. (Three-Phase Circuits) 2. (Magnetic Circuits and Magnetic Materials) 3. (Transformers) 4. (Electromechanical-Energy-Conversion Principles) 5. (Introduction to Rotating Machines) 6. (Synchronous Machines) 7. (Polyphase Induction Machines) 8. (DC Machines) 9. (Single- and Two-Phase Motors)

10. (Introduction to Marine and Aircraft Electrical Systems)

Bibliography

1. Yamayee and Bala, Electromechanical Energy Devices and Power Systems, Wiley, 1994.

2. Ryff, Electric Machinery, 2/e, Prentice Hall, 1994. 3. Sen, Principles of Electric Machines and Power Electronics, 2/e, Wiley, 1997. 4. Cathey, Electric Machines, McGraw-Hill, 2001. 5. Sarma, Electric Machines, 2/e, West, 1994. 6. Wildi, Electrical Machines, Drives, and Power Systems, 5/e, Prentice Hall, 2002. 7. McGeorge, Marine Electrical Equipment and Practice, Newnes, 1993. 8. Pallett, Aircraft Electrical Systems, Wiley, 1987. 9. 1999

10. 1990

1

Chapter 1 Magnetic Circuits and Magnetic Materials � The objective of this course is to study the devices used in the interconversion of electric and

mechanical energy, with emphasis placed on electromagnetic rotating machinery. � The transformer, although not an electromechanical-energy-conversion device, is an important

component of the overall energy-conversion process. � Practically all transformers and electric machinery use ferro-magnetic material for shaping and

directing the magnetic fields that acts as the medium for transferring and converting energy. Permanent-magnet materials are also widely used.

� The ability to analyze and describe systems containing magnetic materials is essential for designing and understanding electromechanical-energy-conversion devices.

� The techniques of magnetic-circuit analysis, which represent algebraic approximations to exact field-theory solutions, are widely used in the study of electromechanical-energy-conversion devices.

§1.1 Introduction to Magnetic Circuits � Assume the frequencies and sizes involved are such that the displacement-current term in

Maxwell’s equations, which accounts for magnetic fields being produced in space by time-varying electric fields and is associated with electromagnetic radiations, can be neglected.

� H : magnetic field intensity, amperes/m, A/m, A-turn/m, A-t/m � B : magnetic flux density, webers/m2, Wb/m2, tesla (T) � 1 Wb = 810 lines (maxwells); 1 T = 410 gauss � From (1.1), we see that the source of H is the current density J . The line integral of the

tangential component of the magnetic field intensity H around a closed contour C is equal to the total current passing through any surface S linking that contour.

�� ⋅=sc

daJHdl (1.1)

� Equation (1.2) states that the magnetic flux density B is conserved. No net flux enters or leaves a closed surface. There exists no monopole charge sources of magnetic fields.

0=⋅�s daB (1.2)

� A magnetic circuit consists of a structure composed for the most part of high-permeability magnetic material. The presence of high-permeability material tends to cause magnetic flux to be confined to the paths defined by the structure.

Figure 1.1 Simple magnetic circuit.

2

� In Fig. 1.1, the source of the magnetic field in the core is the ampere-turn product iN , the magnetomotive force (mmf) F acting on the magnetic circuit.

� The magnetic flux φ (in weber, Wb) crossing a surface S is the surface integral of the normal component B :

� ⋅=s

daBφ (1.3)

� cφ : flux in core, cB : flux density in core

ccc AB=φ (1.4) � cH : average magnitude H in the core. The direction of cH can be found from the RHR.

�== HdlNiF (1.5)

cclHNiF == (1.6) � The relationship between the magnetic field intensity H and the magnetic flux density B :

HB µ= (1.7) � Linear relationship? � 0µµµ r= , µ : magnetic permeability, Wb/A-t-m = H/m

� 70 104 −×= πµ : the permeability of free space

� rµ : relative permeability, typical values: 2000-80,000 � A magnetic circuit with an air gap is shown in Fig. 1.2. Air gaps are present for moving

elements. The air gap length is sufficiently small. φ : the flux in the magnetic circuit.

Figure 1.2 Magnetic circuit with air gap.

cc A

Bφ= (1.8)

gg A

Bφ= (1.9)

ggcc lHlHF += (1.10)

gB

lB

F gc

c

0µµ+= (1.11)

��

�

�

��

�

�+=

gc

c

Ag

Al

F0µµ

φ (1.12)

3

� cR , gR : the reluctance of the core and the air gap, respectively,

c

cc A

lR

µ= ,

gg A

gR

0µ= (1.13), (1.14)

( )gc RRF += φ (1.15)

gc RRF+

=φ (1.16)

gc

c

Ag

Al

F

0µµ

φ+

= (1.17)

� In general, for any magnetic circuit of total reluctance totR , the flux can be found as

totRF=φ (1.18)

� The permeance P is the inverse of the reluctance

tottot R

P1= (1.19)

� Fig. 1.3: Analogy between electric and magnetic circuits:

Figure 1.3 Analogy between electric and magnetic circuits: (a) electric ckt, (b) magnetic ckt.

� Note that with high material permeability: gc RR << and thus gtot RR << ,

g

ANi

g

AF

RF gg

g

00 µµφ ==≈ (1.20)

� Fig. 1.4: Fringing effect, effective gA increased.

Figure 1.4 Air-gap fringing fields.

4

� In general, magnetic circuits can consist of multiple elements in series and parallel.

��� ===k

kkk

k lHFHdlF (1.21)

� ⋅=s

daJF (1.22)

�=k

kk iRV (1.23)

0=�n

ni (1.24)

0=�n

nφ (1.25)

Figure 1.5 Simple synchronous machine.

5

§1.2 Flux Linkage, Inductance, and Energy

� Faraday’s Law:

�� ⋅−=⋅sc

daBdtd

dsE (1.26)

� λ : the flux linkage of the winding, ϕ : the instantaneous value of a time-varying flux, � e : the induced voltage at the winding terminals

dtd

dtd

Neλϕ == (1.27)

ϕλ N= (1.28) � L : the inductance (with material of constant permeability), H = Wb-t/A

iL

λ= (1.29)

totRN

L2

= (1.30)

� The inductance of the winding in Fig. 1.2:

( ) g

AN

AgN

L g

g

02

0

2

/

µµ

== (1.31)

Figure 1.6 (a) Magnetic circuit and (b) equivalent circuit for Example 1.3.

6

7

� Magnetic circuit with more than one windings, Fig. 1.8:

Figure 1.8 Magnetic circuit with two windings.

2211 iNiNF += (1.32)

( )gA

iNiN c02211

µφ += (1.33)

20

21102

111 igA

NNigA

NN cc���

����

�+��

�

����

�==

µµφλ (1.34)

2121111 iLiL +=λ (1.35)

gA

NL c02111

µ= (1.36)

210

2112 LgA

NNL c ==µ

(1.37)

202

210

2122 igA

NigA

NNN cc���

����

�+��

�

����

�==

µµφλ (1.38)

2221212 iLiL +=λ (1.39)

gA

NL c02222

µ= (1.40)

� Induced voltage, power (W = J/s), and stored energy:

( )Lidtd

e = (1.41)

( )Lidtd

Le = (1.42)

dtdL

idtdi

Le += (1.43)

dtd

iiepλ== (1.44)

�� ==∆ 2

1

2

1

λ

λλdidtpW

t

t (1.45)

( )�� −===∆ 2

1

2

1

21

222

1λ

λ

λ

λλλλλλ

Ld

LdiW (1.46)

22

221

iL

LW == λ (1.47)

8

§1.3 Properties of Magnetic Materials � The importance of magnetic materials is twofold:

� Magnetic materials are used to obtain large magnetic flux densities with relatively low levels of magnetizing force.

� Magnetic materials can be used to constrain and direct magnetic fields in well-defined paths.

� Ferromagnetic materials, typically composed of iron and alloys of iron with cobalt, tungsten, nickel, aluminum, and other metals, are by far the most common magnetic materials. � They are found to be composed of a large number of domains. � When unmagnetized, the domain magnetic moments are randomly oriented. � When an external magnetizing force is applied, the domain magnetic moments tend to

align with the applied magnetic field until all the magnetic moments are aligned with the applied field, and the material is said to be fully saturated.

� When the applied field is reduced to zero, the magnetic dipole moments will no longer be totally random in their orientation and will retain a net magnetization component along the applied field direction.

� The relationship between B and H for a ferromagnetic material is both nonlinear and multivalued. � In general, the characteristics of the material cannot be described analytically but are

commonly presented in graphical form. � The most common used curve is the HB − curve. � Dc or normal magnetization curve: � Hysteresis loop (Note the remanance):

9

Figure 1.9 B-H loops for M-5 grain-oriented electrical steel 0.012 in thick.

Only the top halves of the loops are shown here. (Armco Inc.)

Figure 1.10 Dc magnetization curve for M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.)

Figure 1.13 Hysteresis loop.

10

§1.4 AC Excitation � In ac power systems, the waveforms of voltage and flux closely approximate sinusoidal

functions of time. We are to study the excitation characteristics and losses associated with magnetic materials under steady-state ac operating conditions.

� Assume a sinusoidal variation of the core flux )(tϕ :

( ) tBAtt c ωωφϕ sinsin maxmax == (1.48) where webersinfluxcoreofamplitudemax ϕφ =

teslasindensityfluxofamplitudemax cBB = fπω 2frequencyangular ==

Hzinfrequency=f

� The voltage induced in the N-turn winding is ( ) ( ) tEtNte ωωφω coscos maxmax == (1.49)

maxmaxmax 2 BfNANE cπφω == (1.50)

� The Root-Mean-Squared (rms) value:

( ) ��

���

�= �T

odttf

TF 2

rms

1 (1.51)

maxmaxrms 22

2BfNABfNAE cc ππ == (1.52)

Note that the rums value of a sinusoidal wave is 21 times its peak value.

� Excitation phenomena, Fig. 1.11:

� cc HBi vs vs ⇔ϕϕ , ϕi : exciting current.

� Note that cc AB=ϕ and that NHi cc /λ=ϕ .

11

Figure 1.11 Excitation phenomena. (a) Voltage, flux, and exciting current;

(b) corresponding hysteresis loop.

N

HII rmscc

rms,

, =ϕ (1.53)

( )ccrms

rmsccrmsrms

lAHfNBNHI

BfNAIE

max

max,

2

2

π

πϕ

=

= (1.54)

rmsc

rmsrmsa HB

fIEP max

, 2mass ρ

πϕ == (1.55)

aP : the exciting rms voltamperes per unit mass, ccA λc mass ρ=

� The rms exciting voltampere can be seen to be a property of the material alone. It depends only on maxB because rmsH is a unique function of maxB .

Figure 1.12 Exciting rms voltamperes per kilogram at 60 Hz for

M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.) � The exciting current supplies the mmf required to produce the core flux and the power input

associated with the energy in the magnetic field in the core. � Part of this energy is dissipated as losses and results in heating of the core. � The rest appears as reactive power associated with energy storage in the magnetic field.

This reactive power is not dissipated in the core; it is cyclically supplied and absorbed by the excitation source.

12

� Two loss mechanisms are associated with time-varying fluxes in magnetic materials. � The first is ohmic RI 2 heating, associated with induced currents in the core material.

� Eddy currents circulate and oppose changes in flux density in the material. � To reduce the effects, magnetic structures are usually built of thin sheets of

laminations of the magnetic material. � Eddy-current loss ∝ 2f , 2

maxB � The second loss mechanic is due to the hysteretic nature of magnetic material.

� The energy input W to the core as the material undergoes a single cycle

( )� �� =��

���

�== cccccccc dBHlANdBA

NlH

diW λϕ (1.56)

� For a given flux level, the corresponding hysteresis losses are proportional to the area of the hysteresis loop and to the total volume of material.

� Hysteresis power loss ∝ f � Information on core loss is typically presented in graphical form. It is plotted in terms of

watts per unit weight as a function of flux density; often a family of curves for different frequencies are given. See Fig. 1.14.

Figure 1.13 Hysteresis loop; hysteresis loss is proportional to the loop area (shaded).

Figure 1.14 Core loss at 60 Hz in watts per kilogram for

M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc).

13

Figure 1.15 Laminated steel core with winding for Example 1.8.

14

§1.5 Permanent Magnets � Certain magnetic materials, commonly known as permanent-magnet materials, are

characterized by large values of remanent magnetization and coercivity. These materials produce significant magnetic flux even in magnetic circuits with air gaps.

� The second quadrant of a hysteresis loop (the magnetization curve) is usually employed for analyzing a permanent-magnet material. � rB : residual flux density or remanent magnetization, � cH : coercivity, (1) a measure of the magnitude of the mmf required to demagnetize the

material, and (2) a measure of the capability of the material to produce flux in a magnetic circuit which includes an air gap.

� Large value (> 1 kA/m): hard magnetic material, o.w.: soft magnetic material � Fig. 1.16(a): Alnico 5, rB T 22.1≅ , cH kA/m 49−≅ � Fig. 1.16(b): M-5 steel, rB T 4.1≅ , cH kA/m 6−≅ � Both Alnico 5 and M-5 electrical steel would be useful in producing flux in unexcited

magnetic circuits since they both have large values of remanent magnetization. � The significant of remanent magnetization is that it can produce magnetic flux in a magnetic

circuit in the absence of external excitation (such as winding currents).

Figure 1.16 (a) Second quadrant of hysteresis loop for Alnico 5; (b) second quadrant of hysteresis loop for

M-5 electrical steel; (c) hysteresis loop for M-5 electrical steel expanded for small B. (Armco Inc.)

Figure 1.17 Magnetic circuit for Example 1.9.

15

� Maximum Energy Product: a useful measure of the capability of permanent-magnet material.

� The product of B and H has the dimension of energy density (J/m3) � Choosing a material with the largest available maximum energy product can result in the

smallest required magnet volume. � Consider Example 1.9. From (1.58) and (1.57), (1.59) can be obtained.

mg

mg B

AA

B = , 1−=gHlH

g

mm (1.57) (1.58)

( )

( )mm

mmg

mmg

BH

BHgA

AlB

−��

�

�

��

�

�=

−��

�

�

��

�

�=

gapair

mag0

02

Vol

Volµ

µ

(1.59)

( )mm

g

BH

B

−=

0

2gapair

mag

VolVol

µ (1.60)

� Equation (1.60) indicates that to achieve a desired flux density in the air gap the required volume of the magnet can be minimized by operating the magnet at the point of maximum energy product.

� A curve of constant B-H product is a hyperbola. � In Fig. 1.16a, the maximum energy product for Alnico 5 is 40 kJ/m3, occurring at the point

T 0.1=B and kA/m 40−=H .

16

Figure 1.18 Magnetic circuit for Example 1.10.

§1.6 Application of Permanent Magnet Materials

Figure 1.19 Magnetization curves for common permanent-magnet materials.

1

Chapter 2 Transformers � This chapter is to discuss certain aspects of the theory of magnetically-coupled circuits, with

emphasis on transformer action. � The static transformer is not an energy conversion device, but an indispensable component in

many energy conversion systems. � It is a significant component in ac power systems:

� Electric generation at the most economical generator voltage � Power transfer at the most economical transmission voltage � Power utilization at the most voltage for the particular utilization device

� It is widely used in low-power, low-current electronic and control circuits: � Matching the impedances of a source and its load for maximum power transfer � Isolating one circuit from another � Isolating direct current while maintaining ac continuity between two circuits

� The transformer is one of the simpler devices comprising two or more electric circuits coupled by a common magnetic circuit. � Its analysis involves many of the principles essential to the study of electric machinery.

§2.1 Introduction to Transformers � Essentially, a transformer consists of two or more windings coupled by mutual magnetic flux.

� One of these windings, the primary, is connected to an alternating-voltage. � An alternating flux will be produced whose magnitude will depend on the primary voltage,

the frequency of the applied voltage, and the number of turns. � The mutual flux will link the other winding, the secondary, and will induce a voltage in it

whose value will depend on the number of secondary turns as well as the magnitude of the mutual flux and the frequency.

� By properly proportioning the number of primary and secondary turns, almost any desired voltage ratio, or ratio of transformation, can be obtained.

� The essence of transformer action requires only the existence of time-varying mutual flux linking two windings. � Iron-core transformer: coupling between the windings can be made much more effectively

using a core of iron or other ferromagnetic material. � The magnetic circuit usually consists of a stack of thin laminations. � Silicon steel has the desirable properties of low cost, low core loss, and high permeability

at high flux densities (1.0 to 1.5 T). � Silicon-steel laminations 0.014 in thick are generally used for transformers operating

at frequencies below a few hundred hertz. � Two common types of construction: core type and shell type (Fig. 2.1).

Figure 2.1 Schematic views of (a) core-type and (b) shell-type transformers.

2

� Most of the flux is confined to the core and therefore links both windings. � Leakage flux links one winding without linking the other. � Leakage flux is a small fraction of the total flux. � Leakage flux is reduced by subdividing the windings into sections and by placing

them as close together as possible.

§2.2 No-Load Conditions � Figure 2.4 shows in schematic form a transformer with its secondary circuit open and an

alternating voltage 1v applied to its primary terminals.

Figure 2.4 Transformer with open secondary.

� The primary and secondary windings are actually interleaved in practice. � A small steady-state current ϕi (the exciting current) flows in the primary and establishes

an alternating flux in the magnetic current. � 1e = emf induced in the primary (counter emf)

1λ = flux linkage of the primary winding ϕ = flux in the core linking both windings

1N = number of turns in the primary winding � The induced emf (counter emf) leads the flux by ο90 .

dtd

Ndt

de

ϕλ1

11 == (2.1)

111 eiRv += ϕ (2.2)

� 11 ve ≈ if the no-load resistance drop is very small and the waveforms of voltage and flux are very nearly sinusoidal.

tωφϕ sinmax= (2.3)

tdtd

Ne ωωφϕcosmax11 == (2.4)

max1max11 22

2 φπφπNfNfE == (2.5)

1

1max

2 Nf

V

πφ = (2.6)

� The core flux is determined by the applied voltage, its frequency, and the number of turns

3

in the winding. The core flux is fixed by the applied voltage, and the required exciting current is determined by the magnetic properties of the core; the exciting current must adjust itself so as to produce the mmf required to create the flux demanded by (2.6).

� A curve of the exciting current as a function of time can be found graphically from the ac hysteresis loop as shown in Fig. 1.11.

Figure 1.11 Excitation phenomena. (a) Voltage, flux, and exciting current; (b) corresponding hysteresis loop.

� If the exciting current is analyzed by Fourier-series methods, its is found to consist of a

fundamental component and a series of odd harmonics. � The fundamental component can, in turn, be resolved into two components, one in phase

with the counter emf and the other lagging the counter emf by ο90 . � Core-loss component: the in-phase component supplies the power absorbed by

hysteresis and eddy-current losses in the core. � Magnetizing current: It comprises a fundamental component lagging the counter emf by

ο90 , together with all the harmonics, of which the principal is the third (typically 40%). � The peculiarities of the exciting-current waveform usually need not by taken into account,

because the exciting current itself is small, especially in large transformers. (typically about 1 to 2 percent of full-load current)

� Phasor diagram in Fig. 2.5.

1E = the rms value of the induced emf

Φ = the rms value of the flux

ϕI = the rms value of the equivalent sinusoidal exciting current

� ϕI lags 1E by a phase angle cθ .

Figure 2.5 No-load phasor diagram.

4

� The core loss cP equals the product of the in-phase components of the 1E and ϕI :

cc IEP θϕ cos1= (2.7)

� cI = core-loss current, mI = magnetizing current

§2.3 Effect of Secondary Current; Ideal Transformer

Figure 2.6 Ideal transformer and load.

� Ideal Transformer (Fig. 2.6) � Assumptions:

1. Winding resistances are negligible. 2. Leakage flux is assumed negligible. 3. There are no losses in the core. 4. Only a negligible mmf is required to establish the flux in the core.

� The impressed voltage, the counter emf, the induced emf, and the terminal voltage:

dtd

Nevϕ

111 == , dtd

Nevϕ

222 == (2.8)(2.9)

2

1

2

1

NN

vv

= (2.10)

� An ideal transformer transforms voltages in the direct ratio of the turns in its windings. � Let a load be connected to the secondary.

02211 =− iNiN , 2211 iNiN = (2.11)(2.12)

1

2

2

1

NN

ii

= (2.13)

� An ideal transformer transforms currents in the inverse ratio of the turns in its windings.

5

� From (2.10) and (2.13), 2211 iviv = (2.14)

� Instantaneous power input to the primary equals the instantaneous power output from the secondary.

� Impedance transformation properties: Fig. 2.7.

Errore. Figure 2.7 Three circuits which are identical at terminals ab when the transformer is ideal.

22

11 ˆˆ v

NN

v = and 11

22 ˆˆ v

NN

v = (2.15)

22

11

ˆˆ INN

I = and 11

22

ˆˆ INN

I = (2.16)

2

2

2

2

1

1

1

ˆ

ˆ

ˆ

ˆ

I

VNN

I

V���

����

�= (2.17)

2

22 ˆ

ˆ

I

VZ = (2.18)

2

2

2

11 Z

NN

Z ���

����

�= (2.19)

� Transferring an impedance from one side to the other is called “referring the impedance

to the other side.” Impedances transform as the square of the turns ratio.

� Summary for the ideal transformer: � Voltages are transformed in the direct ratio of turns. � Currents are transformed in the inverse ratio of turns. � Impedances are transformed in the direct ratio squared. � Power and voltamperes are unchanged.

6

§2.4 Transformer Reactances and Equivalent Circuits � A more complete model must take into account the effects of winding resistances, leakage

fluxes, and finite exciting current due to the finite and nonlinear permeability of the core. � Note that the capacitances of the windings will be neglected. � Method of the equivalent circuit technique is adopted for analysis.

� Development of the transformer equivalent circuit � Leakage flux: Fig. 2.9.

Figure 2.9 Schematic view of mutual and leakage fluxes in a transformer.

� 11L = primary leakage inductance,

11X = primary leakage reactance

11 11 2 LfX π= (2.20)

� Effect of the primary winding resistance: 1R � Effect of the exciting current:

( ) 2221

22111

ˆˆˆ

ˆˆˆ

INIIN

INININ

−′+=

−=

ϕ

ϕ (2.21)

21

22

ˆˆ INN

I =′ (2.22)

� mL = magnetizing inductance, mX = magnetizing reactance

mm LfX π2= (2.23)

7

� Ideal transformer:

2

1

2

1

ˆ

ˆ

NN

E

E= (2.24)

� Secondary resistance, secondary leakage reactance � Equivalent-T circuit for a transformer:

22 1

2

2

11

ˆ XNN

X ���

����

�= , 2

2

2

12 R

NN

R ���

����

�=′ , 2

2

2

12 V

NN

V ���

����

�=′ (2.25)-(2.27)

� Steps in the development of the transformer equivalent circuit: Fig. 2.10.

� The actual transformer can be seen to be equivalent to an ideal transformer plus external impedances

� Refer to the assumptions for an ideal transformer to understand the definitions and meanings of these resistances and reactances.

Errore.

Figure 2.10 Steps in the development of the transformer equivalent circuit.

8

Figure 2.11 Equivalent circuits for transformer of Example 2.3 referred to (a) the high-voltage side and (b) the low-voltage side.

§2.5 Engineering Aspects of Transformer Analysis � Approximate forms of the equivalent circuit:

Figure 2.12 Approximate transformer equivalent circuits.

9

Figure 2.13 Cantilever equivalent circuit for Example 2.4.

10

Figure 2.14 (a) Equivalent circuit and (b) phasor diagram for Example 2.5.

� Two tests serve to determine the parameters of the equivalent circuits of Figs. 2.10 and 2.12.

� Short-circuit test and open-circuit test � Short-Circuit Test

� The test is used to find the equivalent series impedance eqeq jXR + . � The high voltage side is usually taken as the primary to which voltage is applied. � The short circuit is applied to the secondary � Typically an applied voltage on the order of 10 to 15 % or less of the rated value will result

in rated current. � See Fig. 2.15. Note that mc jXRZ //=ϕ .

Figure 2.15 Equivalent circuit with short-circuited secondary. (a) Complete equivalent circuit.

(b) Cantilever equivalent circuit with the exciting branch at the transformer secondary.

( )2

2

1

12

1211 jXRZ

jXRZjXRZ sc ++

+++=

ϕ

ϕ (2.28)

eqeqsc jXRjXRjXRZ +=+++≈21 1211 (2.29)

� Typically the instrumentation will measure the rms magnitude of the applied voltage scV , the short-circuit current scI , and the power scP . The circuit parameters (referred to the primary) can be found as (2.30)-(2.32).

sc

scsceq I

VZZ == |||| (2.30)

11

2sc

scsceq I

PRR == (2.31)

22|| scscsceq RZXX −== (2.32) � The equivalent impedance can be referred from one side to the other. � Approximate values of the individual primary and secondary resistances and leakage

reactances can be obtained by assuming that eqRRR 5.021 == and eqll XXX 5.021

== when all impedances are referred to the same side.

� Note that it is possible to measure 1R and 2R directly by a dc resistance measurement on each winding. However, no such simple test exists for

1lX and

2lX .

� Open-Circuit Test

� The test is used to find the equivalent shunt impedance mc jXR // . � The test is performed with the secondary open-circuited and rated voltage impressed on the

primary. If the transformer is to be used at other than its rated voltage, the test should be done at that voltage.

� An exciting current of a few percent of full-load current is obtained. � See Fig. 2.16. Note that mc jXRZ //=ϕ .

Figure 2.16 Equivalent circuit with open-circuited secondary. (a) Complete equivalent circuit.

(b) Cantilever equivalent circuit with the exciting branch at the transformer primary. ( )

mc

mcoc jXR

jXRjXRZjXRZ

+++=++=

11 1111 ϕ (2.33)

( )mc

mcoc jXR

jXRZZ

+=≈ ϕ (2.34)

� Typically the instrumentation will measure the rms magnitude of the applied voltage ocV , the open-circuit current ocI , and the power ocP . The circuit parameters (referred to the primary) can be found as (2.35)-(2.37).

oc

occ P

VR

2

= (2.35)

oc

oc

PV

Z =|| ϕ (2.36)

( ) ( )22 /1||/1

1

c

mRZ

X−

=ϕ

(2.37)

� The open-circuit test can be used to obtain the core loss for efficiency computations and to check the magnitude of the exciting current.

� Note the term “Voltage Regulation” which is to be discussed in Example 2.6.

12

13

§2.6 Autotransformers; Multiwinding Transformers � Two-winding � Other winding configurations. §2.6.1 Autotransformers � Autotransformer connection: Fig. 2.17.

Figure 2.17 (a) Two-winding transformer. (b) Connection as an autotransformer.

� The windings of the two-winding transformer are electrically isolated whereas those of the

autotransformer are connected directly together. � In the transformer connection, winding ab must be provided with extra insulation. � Autotransformer have lower leakage reactances, lower losses, and smaller exciting current

and cost less than two-winding transformers when the voltage ration does not differ too greatly from 1:1.

� The rated voltages of the transformer can be expressed in terms of those of the two-winding transformer as

ratedratedVVL 1= (2.38)

ratedratedratedrated LH VN

NNVVV ��

�

����

� +=+=

1

2121 (2.39)

� The effective turns ratio of the autotransformer is thus 121 /)( NNN + . � The power rating of the autotransformer is equal to 221 /)( NNN + times that of the

two-winding transformer.

14

Figure 2.18 (a) Autotransformer connection for Example 2.7.

(b) Currents under rated load.

15

§2.6.2 Multiwinding Transformers � Transformers having three or more windings, known as multiwinding or multicircuit

transformers, are often used to interconnect three or more circuits which may have different voltages. � Trsansformers having a primary and multiple secondaries are frequently found in

multiple-output dc power supplies. � Distribution transformers used to supply power for domestic purposes usually have two

120-V secondaries connected in series. � The three-phase transformer banks used to interconnect two transmission system of

different voltages often have a third, or tertiary, set of windings to provide voltage for auxiliary power purposes in substation or to supply a local distribution system. � Static capacitors or synchronous condensers may be connected to the tertiary windings

for power factor correction or voltage regulation. � Sometimes ∆ -connected tertiary windings are put on three-phase banks to provide a

low-impedance path for third harmonic components of the exciting current to reduce third-harmonic components of the neutral voltage.

§2.7 Transformers in Three-Phase Circuits � Three single-phase transformers can be connected to form a three-phase transformer bank in

any of the four ways shown in Fig. 2.19. Note that 21 / NNa = .

Errore.

Figure 2.19 Common three-phase transformer connections; the transformer windings are indicated by the heavy lines.

� The Y-� connection is commonly used in stepping down from a high voltage to a medium

or low voltage. � The �-Y connection is commonly used for stepping up to a high voltage. � The �-� connection has the advantage that one transformer can be removed for repair or

maintenance while the remaining two continue to function as a three-phase bank with the

16

rating reduced to 58 percent of that of the original bank. (Open-delta, or V, connection) � The Y-Y connection is seldom used because of difficulties with exciting-current

phenomenon. � Because there is no neutral connection to carry harmonics of the exciting current and

harmonic voltages are produced which significantly distort the transformer voltages. � A three-phase bank may consist of one three-phase transformer having all six windings on a

common multi-legged core and contained in a single tank. � They cost less, weigh less, require less floor space, and have somewhat higher efficiency.

� It is usually convenient to carry out circuit computations involving three-phase transformer

banks under balanced conditions on a single-phase (per-phase-Y, line-to-neutral) basis. � Y-�, �-Y, and �-� connections � equivalent Y-Y connections � A balanced �-connected circuit of ∆Z �/phase is equivalent to a balanced Y-connected

circuit of YZ �/phase if

∆= ZZY 31

(2.40)

17

18

§2.8 Voltage and Current Transformers � Transformers are often used in instrumentation applications to match the magnitude of a

voltage or current to the range of a meter or other instrumentation. � Most 60-Hz power-systems’ instrumentation is based upon voltages in the range of 0-120

V rms and currents in the range of 0-5 A rms. � Power system voltages range up to 765-kV line-to-line and currents can be 10’s of kA.

� Some method of supplying an accurate, low-level representation of these signals to the instrumentation is required.

� Potential Transformer (PT) and Current Transformer (CT), also referred to as Instrumentation

Transformer, are designed to approximate the ideal transformer as closely as is practically possible. � The load on an instrumentation transformer is frequently referred to as the burden on that

transformer. � A potential transformer should ideally accurately measure voltage while appearing as an

open circuit to the system under measurement, i.e. drawing negligible current and power. � Its load impedance should be “large” in some sense.

� An ideal current transformer would accurately measure current while appearing as a short circuit to the system under measurement, i.e. developing negligible voltage drop and drawing negligible power.

19

� Its load impedance should be “small” in some sense.

§2.9 The Per-Unit System � Computations relating to machines, transformers, and systems of machines are often carried

out in per-unit system. � All pertinent quantities are expressed as decimal fractions of appropriately chose base

values. � All the usual computations are then carried out in these per unit values instead of the

familiar volts, amperes, ohms, and so on. � Advantages:

� The parameter values typically fall in a reasonably narrow numerical range when expressed in a per-unit system based upon their rating.

� When transformer equivalent-circuit parameters are converted to their per-unit values, the ideal transformer turns ratio becomes 1:1 and hence the ideal transformer can be eliminated.

� Actual quantities: V , I , P ,Q ,VA , R , X , Z ,G , B ,Y

quantityofvalueBasequantityActual

unitperinQuantity = (2.47)

� To a certain extent, base values can be chosen arbitrarily, but certain relations between them must be observed. For a single-phase system:

basebasebasebasebase , , IVVAQP = (2.48)

base

basebasebasebase , ,

IV

ZXR = (2.49)

� Only two independent base quantities can be chose arbitrarily; the remaining quantities are determined by (2.48) and (2.49).

� In typical usage, values of baseVA and baseV are chosen first; values of baseI and all other quantities in (2.48) and (2.49) are then uniquely established.

� The value of baseVA must be the same over the entire system under analysis. � When a transformer is encountered, the values of baseV differ on each side and should

be chosen in the same ratio as the turns ratio of the transformer. � The per-unit ideal transformer will have a unity turns ratio and hence can be

eliminated. � Usually the rated or nominal voltages of the respective sides are chosen.

� The procedure for performing system analyses in per-unit is summarized as follows:

� Machine Ratings as Bases

� When expressed in per-unit form on their rating as a base, the per-unit values of machine

20

parameters fall within a relatively narrow range. � The physics behind each type of device is the same and, in a crude sense, they can

each be considered to be simply scaled versions of the same basic device. � When normalized to their own rating, the effect of the scaling is eliminated and the

result is a set of per-unit parameter values which is quite similar over the whole size range of that device. For power and distribution transformers, pu06.0~02.0=ϕI ,

pu02.0~005.0=R , and pu10.0~015.0=X . � Manufacturers often supply device parameters in per unit on the device base.

� When performing a system analysis, it may be necessary to convert the supplied per-unit parameter values to per-unit values on the base chosen for the analysis.

( ) ( )��

���

=

2base

1base1baseonpu2baseonpu ,,,,

VA

VAVAQPVAQP (2.50)

( ) ( ) ( )( ) �

�

���

=

1base2

2base

2base2

1base1baseonpu2baseonpu ,,,,

VAV

VAVZXPZXP (2.51)

��

���

=

2base

1base1baseonpu2baseonpu V

VVV (2.52)

��

���

=

2base1base

1base2base1baseonpu2baseonpu VAV

VAVII (2.53)

21

Figure 2.22 Transformer equivalent circuits for Example 2.12.

(a) Equivalent circuit in actual units. (b) Per-unit equivalent circuit with 1:1 ideal transformer. (c) Per-unit equivalent circuit following elimination of the ideal transformer.

22

� Balanced Three-Phase System:

� Relations for base values: ( ) phaseper base,phase3basebasebase 3,, VAVAQP =− (2.54)

� The three-phase volt-ampere base ( phase3 base, −VA ) and the line-to-line voltage base

( l-l base,phase3 base, VV =− ) are usually chosen first. � The base values for the phase (line-to-neutral) voltage then is

23

11base,n1base, 31

−− = VV (2.55)

� The base current for three-phase system is equal to the phase current, which is the same as the base current for a single-phase (per-phase) analysis.

phase3base,

phase3base,phaseper base,phase3base,

3 −

−− ==

V

VAII (2.56)

� The three-phase base impedance is chosen to be the single-phase base impedance.

( )phase3base,

2phase3base,

phase3base,

phase3base,

phaseperbase,

n1base,

phaseperbase,phase3base,

3

−

−

−

−

−

−

=

=

=

=

VA

V

I

V

I

V

ZZ

(2.57)

� Note that the factors of 3 and 3 are automatically taken care of in per unit by the base values. Three-phase problems can thus be solved in per unit as if they were single-phase problems.

24

1

Chapter 3 Electromechanical-Energy-Conversion Principles

� The electromechanical-energy-conversion process takes place through the medium of the

electric or magnetic field of the conversion device of which the structures depend on their respective functions. � Transducers: microphone, pickup, sensor, loudspeaker � Force producing devices: solenoid, relay, electromagnet � Continuous energy conversion equipment: motor, generator

� This chapter is devoted to the principles of electromechanical energy conversion and the

analysis of the devices accomplishing this function. Emphasis is placed on the analysis of systems that use magnetic fields as the conversion medium. � The concepts and techniques can be applied to a wide range of engineering situations

involving electromechanical energy conversion. � Based on the energy method, we are to develop expressions for forces and torques in

magnetic-field-based electromechanical systems.

§3.1 Forces and Torques in Magnetic Field Systems � The Lorentz Force Law gives the force F on a particle of charge q in the presence of

electric and magnetic fields. ( )BvEqF ×+= (3.1)

F : newtons, q : coulombs, E : volts/meter, B : telsas, v : meters/second � In a pure electric-field system,

qEF = (3.2) � In pure magnetic-field systems,

( )BvqF ×= (3.3)

Errore.

Figure 3.1 Right-hand rule for ( )BvqF ×= .

� For situations where large numbers of charged particles are in motion,

( )BvEFv ×+= ρ (3.4) vJ ρ= (3.5) BJFv ×= (3.6)

ρ (charge density): coulombs/m3, vF (force density): newtons/m3, J vρ= (current density): amperes/m2.

2

Errore.

Figure 3.2 Single-coil rotor for Example 3.1.

� Unlike the case in Example 3.1, most electromechanical-energy-conversion devices contain

magnetic material. � Forces act directly on the magnetic material of these devices which are constructed of

rigid, nondeforming structures. � The performance of these devices is typically determined by the net force, or torque,

acting on the moving component. It is rarely necessary to calculate the details of the internal force distribution.

� Just as a compass needle tries to align with the earth’s magnetic field, the two sets of fields associated with the rotor and the stator of rotating machinery attempt to align, and torque is associated with their displacement from alignment. � In a motor, the stator magnetic field rotates ahead of that of the rotor, pulling on it

and performing work. � For a generator, the rotor does the work on the stator.

3

� The Energy Method � Based on the principle of conservation of energy: energy is neither created nor destroyed;

it is merely changed in form. � Fig. 3.3(a): a magnetic-field-based electromechanical-energy-conversion device.

� A lossless magnetic-energy-storage system with two terminals � The electric terminal has two terminal variables: e (voltage), i (current). � The mechanical terminal has two terminal variables: fldf (force), x (position) � The loss mechanism is separated from the energy-storage mechanism.

– Electrical losses: ohmic losses… – Mechanical losses: friction, windage…

� Fig. 3.3(b): a simple force-producing device with a single coil forming the electric terminal, and a movable plunger serving as the mechanical terminal. � The interaction between the electric and mechanical terminals, i.e. the

electromechanical energy conversion, occurs through the medium of the magnetic stored energy.

Figure 3.3 (a) Schematic magnetic-field electromechanical-energy-conversion device;

(b) simple force-producing device.

� fldW : the stored energy in the magnetic field

dtdx

feidtWd

fldfld −= (3.7)

dtd

eλ= (3.8)

dxfidWd fldfld −= λ (3.9)

� Equation (3.9) permits us to solve for the force simply as a function of the flux λ and the mechanical terminal position x .

� Equations (3.7) and (3.9) form the basis for the energy method.

4

§3.2 Energy Balance � Consider the electromechanical systems whose predominant energy-storage mechanism is in

magnetic fields. For motor action, we can account for the energy transfer as

���

�

�

���

�

�

+���

�

�

���

�

�

+���

�

�

���

�

�

=���

�

�

���

�

�

heat intoconvertedEnergy

fieldmagneticin stored

energyin Increase

outputenergyMechanical

sourceselectric form

inputEnergy (3.10)

� Note the generator action. � The ability to identify a lossless-energy-storage system is the essence of the energy method.

� This is done mathematically as part of the modeling process. � For the lossless magnetic-energy-storage system of Fig. 3.3(a), rearranging (3.9) in form

of (3.10) gives

fldmechelec dWdWdW += (3.11) where elecdW idλ= = differential electric energy input mech flddW f dx= = differential mechanical energy output flddW = differential change in magnetic stored energy

� Here e is the voltage induced in the electric terminals by the changing magnetic stored energy. It is through this reaction voltage that the external electric circuit supplies power to the coupling magnetic field and hence to the mechanical output terminals.

dteidW =elec (3.12) � The basic energy-conversion process is one involving the coupling field and its action and

reaction on the electric and mechanical systems. � Combining (3.11) and (3.12) results in

fldmechelec dWdWdteidW +== (3.13)

§3.3 Energy in Singly-Excited Magnetic Field Systems � We are to deal energy-conversion systems: the magnetic circuits have air gaps between the

stationary and moving members in which considerable energy is stored in the magnetic field. � This field acts as the energy-conversion medium, and its energy is the reservoir between

the electric and mechanical system. � Fig. 3.4 shows an electromagnetic relay schematically. The predominant energy storage

occurs in the air gap, and the properties of the magnetic circuit are determined by the dimensions of the air gap.

Errore. Figure 3.4 Schematic of an electromagnetic relay.

5

( )ixL=λ (3.14) dxfdW fldmech = (3.15)

dxfiddW fldfld −= λ (3.16)

� fldW is uniquely specified by the values of λ and x . Therefore, λ and x are referred to as state variables.

� Since the magnetic energy storage system is lossless, it is a conservative system. fldW is the same regardless of how λ and x are brought to their final values. See Fig. 3.5 where tow separate paths are shown.

Figure 3.5 Integration paths for fldW .

( ) �� +=2bpath

fld2apath

fld00fld , dWdWxW λ (3.17)

On path 2a, 0dλ = and fld 0f = . Thus, fld 0dW = on path 2a. On path 2b, 0dx = . Therefore, (3.17) reduces to the integral of idλ over path 2b.

( ) ( ) λλλλ

dxixW �= 0

0 000fld ,, (3.18)

For a linear system in which λ is proportional to i , (3.18) gives

( ) ( ) ( ) ( )�� =′′=′′=

λλ λλλλλλ0

2

0fld 21

,,xL

dxL

dxixW (3.19)

� V : the volume of the magnetic field

( )fld 0

B

VW H dB dV′= ⋅� � (3.20)

If B Hµ= , 2

fld 2V

BW dV

µ� �

= � �� �

� (3.21)

6

Figure 3.6 (a) Relay with movable plunger for Example 3.2. (b) Detail showing air-gap configuration with the plunger partially removed.

Errore.

7

§3.4 Determination of Magnetic Force and Torque form Energy � The magnetic stored energy fldW is a state function, determined uniquely by the values of the

independent state variables λ and x .

( ) dxfidxdW fldfld , −= λλ (3.22)

( )2 1

1 2 1 21 2

,x x

F FdF x x dx dx

x x∂ ∂= +∂ ∂

(3.23)

( ) dxx

Wd

WxdW

x λ

λλ

λ∂

∂+

∂∂

= fldfldfld , (3.24)

Comparing (3.22) with (3.24) gives (3.25) and (3.26):

( )x

xWi

λλ

∂∂

=,fld (3.25)

( )λ

λx

xWf

∂∂

−=,fld

fld (3.26)

� Once we know fldW as a function of λ and x , (3.25) can be used to solve for ( , )i xλ . � Equation (3.26) can be used to solve for the mechanical force fld ( , )f xλ . The partial

derivative is taken while holding the flux linkages λ constant. � For linear magnetic systems for which ( )L x iλ = , the force can be found as

( ) ( )( )

dxxdL

xLxLxf 2

22

fld22

1 λλ

λ

=���

����

�

∂∂−= (3.27)

( )dx

xdLif

2

2

fld = (3.28)

8

Figure 3.7 Example 3.3. (a) Polynomial curve fit of inductance. (b) Force as a function of position x for i = 0.75 A.

� For a system with a rotating mechanical terminal, the mechanical terminal variables become

the angular displacement θ and the torque fldT . ( ) θλθλ dTiddW fldfld , −= (3.29)

( )λθ

θλ∂

∂−=

,fldfld

WT (3.30)

� For linear magnetic systems for which ( )L iλ θ= :

( ) ( )θλθλ

LW

2

fld 21

, = (3.31)

( ) ( )( )θθ

θλ

θλ

θλ

ddL

LLT 2

22

fld 21

21 =��

�

����

�

∂∂−= (3.32)

(3.33) ( )θθ

ddLi

T2

2

fld = (3.34)

Figure 3.9 Magnetic circuit for Example 3.4.

9

§3.5 Determination of Magnetic Force and Torque from Coenergy � Recall that in §3.4, the magnetic stored energy fldW is a state function, determined uniquely

by the values of the independent state variables λ and x . ( ) dxfidxdW fldfld , −= λλ (3.22)

( ) dxx

Wd

WxdW

x λ

λλ

λ∂

∂+

∂∂

= fldfldfld , (3.24)

( )x

xWi

λλ

∂∂

=,fld (3.25)

( )λ

λx

xWf

∂∂

−=,fld

fld (3.26)

� Coenergy: from which the force can be obtained directly as a function of the current. The

selection of energy or coenergy as the state function is purely a matter of convenience. � The coenergy ),( xiW fld′ is defined as a function of i and x such that

( ) ( )xWixiW ,, fldfld λλ −=′ (3.34) ( ) diidid λλλ += (3.35)

( ) ),()(,fld xdWidxiWd fld λλ −=′ (3.36)

( ) dxfdixiWd fldfld , +=′ λ (3.37)

� From (3.37), the coenergy ),( xiW fld′ can be seen to be a state function of the two

independent variables i and x .

( ) dxx

Wdi

i

WxiWd

i

fld

x

fld

∂

′∂+

∂

′∂=′ ,fld (3.38)

( )xi

xiW∂′∂

=,fldλ (3.39)

( )ix

xiWf

∂′∂

=,fld

fld (3.40)

� For any given system, (3.26) and (3.40) will give the same result.

10

� By analogy to (3.18) in §3.3, the coenergy can be found as (3.41)

( ) ( ) λλλλ

dxixW �= 0

0 000fld ,, (3.18)

( ) ( )� ′′=′i

idxixiW0fld ,, λ (3.41)

For linear magnetic systems for which ixL )(=λ ,

( ) ( ) 2fld 2

1, ixLxiW =′ (3.42)

( )dx

xdLif

2

2

fld = (3.43)

� (3.43) is identical to the expression given by (3.28). � For a system with a rotating mechanical displacement,

( ) ( ) idiiWi

′′=′ �0fld ,, θλθ (3.44)

( )i

iWT

θθ

∂′∂

=,fld

fld (3.45)

If the system is magnetically linear,

( ) ( ) 2fld 2

1, iLiW θθ =′ (3.46)

( )θθ

ddLi

T2

2

fld = (3.47)

� (3.47) is identical to the expression given by (3.33). � In field-theory terms, for soft magnetic materials

� � ����

�� ⋅=′

V

HdVdHBW

0

0fld (3.48)

dVH

Wv�=′

2

2

fld

µ (3.49)

For permanent-magnet (hard) materials

� � ����

�� ⋅=′

V

H

HdVdHBW

c

0

fld (3.50)

11

� For a magnetically-linear system, the energy and coenergy (densities) are numerically equal:

22

21

/21

LiL =λ , 22

21

/21

HB µµ = . For a nonlinear system in which λ and i or B and

H are not linearly proportional, the two functions are not even numerically equal. iWW λ=′+ fldfld (3.51)

Figure 3.10 Graphical interpretation of energy and coenergy in a singly-excited system.

� Consider the relay in Fig. 3.4. Assume the relay armature is at position x so that the

device operating at point a in Fig. 3.11. Note that

( )λλ

λxW

xxW

fx ∆

∆−≅

∂∂

−=→∆

fld

0

fldfld lim

, and

( )i

xi x

Wx

xiWf

∆′∆

≅∂′∂

=→∆

fld

0

fldfld lim

,

Figure 3.11 Effect of ∆x on the energy and coenergy of a singly-excited device:

(a) change of energy with λ held constant; (b) change of coenergy with i held constant.

12

� The force acts in a direction to decrease the magnetic field stored energy at constant flux or to increase the coenergy at constant current. � In a singly-excited device, the force acts to increase the inductance by pulling on

members so as to reduce the reluctance of the magnetic path linking the winding.

Figure 3.12 Magnetic system of Example 3.6.

13

§3.6 Multiply-Excited Magnetic Field Systems � Many electromechanical devices have multiple electrical terminals.

� Measurement systems: torque proportional to two electric signals; power as the product of voltage and current.

� Energy conversion devices: multiply-excited magnetic field system. � A simple system with two electrical terminals and one mechanical terminal: Fig. 3.13.

� Three independent variables: },,{ 21 λλθ , },,{ 21 iiθ , },,{ 21 iλθ , or },,{ 21 λθ i . ( ) θλλθλλ dTdididW fld221121fld ,, −+= (3.52)

Figure 3.13 Multiply-excited magnetic energy storage system.

( )

θλλ

θλλ

,1

21fld1

2

,,

∂∂

=W

i (3.53)

( )θλ

λθλλ

,2

21fld2

1

,,

∂∂

=W

i (3.54)

( )21 ,

21fldfld

,,

λλθ

θλλ∂

∂−=

WT (3.55)

To find fldW , use the path of integration in Fig. 3.14.

( ) ( ) ( ) 102210 120210 2021fld ,,,,0,,0

0102

00λθθλλλλθθλλθλλ

λλdidiW ==+=== �� (3.56)

Figure 3.14 Integration path to obtain ( )021fld ,,

00θλλW .

14

� In a magnetically-linear system,

2121111 iLiL +=λ (3.57)

2221212 iLiL +=λ (3.58)

2112 LL = (3.59)

Note that )(θijij LL = .

D

LLi 212122

1

λλ −= (3.60)

D

LLi 211121

2

λλ +−= (3.61)

21122211 LLLLD −= (3.62) The energy for this linear system is

( ) ( )( )

( ) ( )( )( )

( ) ( ) ( ) ( ) ( )( ) 0000

01 002

00

210

01221022

0

22011

0

0 10

20121022

0 20

2011021fld

21

21

,,

λλθθ

λθθ

λθθ

λθ

λθλθλ

θλθ

θλλλλ

D

LL

DL

D

dD

LLd

D

LW

−+=

−+= ��

(3.63)

� Coenergy function for a system with two windings can be defined as (3.46) ( ) fld221121fld ,, WiiiiW −+=′ λλθ (3.64)

( ) θλλθ dTdidiiiWd fld221121fld ,, ++=′ (3.65)

( )θ

θλ

,1

21fld1

2

,,

ii

iiW

∂∂

= (3.66)

( )θ

θλ

,2

21fld2

1

,,

ii

iiW

∂∂

= (3.67)

( )21 ,

21fldfld

,,

ii

iiWT

θθ

∂

′∂= (3.68)

( ) ( ) ( ) 102210 120210 2021fld ,,,,0,,0

0102diiiidiiiiiW

iθθλθθλθ

λ==+===′ �� (3.69)

� For the linear system described as (3.57) to (3.59)

( ) ( ) ( ) ( ) 21122222

2111021fld 2

121

,, iiLiLiLiiW θθθθ ++=′ (3.70)

( ) ( ) ( ) ( )θ

θθ

θθ

θθ

θd

dLii

ddLi

ddLiiiW

Tii

1221

222211

21

,

021fldfld 22

,,

21

++=∂

′∂= (3.71)

� Note that (3.70) is simpler than (3.63). That is, the coenergy function is a relatively simple function of displacement.

� The use of a coenergy function of the terminal currents simplifies the determination of torque or force.

� Systems with more than two electrical terminals are handled in analogous fashion.

15

Figure 3.15 Multiply-excited magnetic system for Example 3.7.

Figure 3.16 Plot of torque components for the multiply-excited system of Example 3.7.

16

Practice Problem 3.7

Find an expression for the torque of a symmetrical two-winding system whose inductances vary as

θ4cos27.08.02211 +== LL θ2cos65.012 =L

for the condition that A37.021 =−= ii . Solution: θθ 2sin178.04sin296.0fld +−=T

___________________________________________________________________

� System with linear displacement:

( ) ( ) ( ) 102210 120210 2021fld ,,,,0,,0

0102

00λλλλλλλλλ

λλdxxidxxixW ==+=== �� (3.72)

( ) ( ) ( ) 102210 120210 2021fld ,,,,0,,0

0102

00dixxiiidixxiixiiW ==+===′ ��

λλλλ (3.73)

( )21 ,

21fldfld

,,

λλ

λλx

xWf

∂∂

−= (3.74)

( )21 ,

21fldfld

,,

iix

xiiWf

∂

′∂−= (3.75)

For a magnetically-linear system,

( ) ( ) ( ) ( ) 21122222

211121fld 2

121

,, iixLixLixLxiiW ++=′ (3.76)

( ) ( ) ( )dx

xdLii

dxxdLi

dxxdLi

f 1221

222211

21

fld 22++= (3.77)

1

Chapter 4 Introduction to Rotating Machines � The objective of this chapter is to introduce and discuss some of the principles underlying the

performance of electric machinery, both ac and dc machines.

§4.1 Elementary Concepts � Voltages can be induced by time-varying magnetic fields. In rotating machines, voltages are

generated in windings or groups of coils by rotating these windings mechanically through a magnetic field, by mechanically rotating a magnetic field past the winding, or by designing the magnetic circuit so that the reluctance varies with rotation of the rotor. � The flux linking a specific coil is changed cyclically, and a time-varying voltage is

generated. � Electromagnetic energy conversion occurs when changes in the flux linkage result from

mechanical motion. � A set of such coils connected together is typically referred to as an armature winding, a

winding or a set of windings carrying ac currents. � In ac machines such as synchronous or induction machines, the armature winding is

typically on the stator. (the stator winding) � In dc machines, the armature winding is found on the rotor. (the rotor winding)

� Synchronous and dc machines typically include a second winding (or set of windings), referred to as the field winding, which carrys dc current and which are used to produce the main operating flux in the machine. � In dc machines, the field winding is found on the stator. � In synchronous machines, the field winding is found on the rotor. � Permanent magnets can be used in the place of field windings.

� In most rotating machines, the stator and rotor are made of electrical steel, and the windings are installed in slots on these structures. The stator and rotor structures are typically built from thin laminations of electrical steel, insulated from each other, to reduce eddy-current losses.

§4.2 Introduction to AC And DC Machines §4.2.1 AC Machines � Traditional ac machines fall into one of two categories: synchronous and induction.

� In synchronous machines, rotor-winding currents are supplied directly from the stationary frame through a rotating contact.

� In induction machines, rotor currents are induced in the rotor windings by a combination of the time-variation of the stator currents and the motion of the rotor relative to the stator.

� Synchronous Machines � Fig. 4.4: a simplified salient-pole ac synchronous generator with two poles.

� The armature winding is on the stator, and the field winding is on the rotor. � The field winding is excited by direct current conducted to it by means of stationary

carbon brushes that contact rotating slip rings or collector rings. � It is advantages to have the single, low-power field winding on the rotor while having

the high-power, typically multiple-phase, armature winding on the stator. � Armature winding ),( aa − consists of a single coil of N turns. � Conductors forming these coil sides are connected in series by end connections. � The rotor is turned at a constant speed by a source of mechanical power connected to its

shaft. Flux paths are shown schematically by dashed lines.

2

Figure 4.4 Schematic view of a simple, two-pole, single-phase synchronous generator.

� Assume a sinusoidal distribution of magnetic flux in the air gap of the machine in Fig. 4.4. � The radial distribution of air-gap flux density B is shown in Fig. 4.5(a) as a function

of the spatial angle θ around the rotor periphery. � As the rotor rotates, the flux –linkages of the armature winding change with time and

the resulting coil voltage will be sinusoidal in time as shown in Fig 4.5(b). The frequency in cycles per second (Hz) is the same as the speed of the rotor in revolutions in second (rps).

� A two-pole synchronous machine must revolve at 3600 rpm to produce a 60-Hz voltage. � Note the terms “rpm” and “rps”.

Figure 4.5 (a) Space distribution of flux density and (b) corresponding waveform of the generated voltage for the single-phase generator of Fig. 4.4.

� A great many synchronous machines have more than two poles. Fig 4.6 shows in

schematic form a four-pole single-phase generator. � The field coils are connected so that the poles are of alternate polarity. � The armature winding consists of two coils ),( 11 aa − and ),( 22 aa − connected in

series by their end connections. � There are two complete wavelengths, or cycles, in the flux distribution around the

periphery, as shown in Fig. 4.7. � The generated voltage goes through two complete cycles per revolution of the rotor. � The frequency in Hz is thus twice the speed in rps.

3

Figure 4.6 Schematic view of a simple, four-pole, single-phase synchronous generator.

Figure 4.7 Space distribution of the air-gap flux density in an idealized,

four-pole synchronous generator.

� When a machine has more than two poles, it is convenient to concentrate on a single pair of poles and to express angles in electrical degrees or electrical radians rather than in physical units. � One pair of poles equals 360 electrical degrees or 2� electrical radians. � Since there are poles/2 wavelengths, or cycles, in one revolution, it follows that

aea 2poles θθ �

�

���

�= (4.1)

where eaθ is the angle in electrical units and aθ is the spatial angle. � The coil voltage of a multipole machine passes through a complete cycle every time a

pair of poles sweeps by, or (poles/2) times each revolution. The electrical frequency

ef of the voltage generated is therefore

Hz602

polese

nf �

�

���

�= (4.2)

where n is the mechanical speed in rpm. Note that me )poles/2( ωω = . � The rotors shown in Figs. 4.4 and 4.6 have salient, or projecting, poles with concentrated

windings. Fig. 4.8 shows diagrammatically a nonsalient-pole, or cylindrical, rotor. � The field winding is a two-pole distributed winding; the coil sides are distributed in

multiple slots around the rotor periphery and arranged to produce an approximately sinusoidal distribution of radial air-gap flux.

� Most power systems in the world operate at frequencies of either 50 or 60 Hz. � A salient-pole construction is characteristic of hydroelectric generators because

hydraulic turbines operate at relatively low speeds, and hence a relatively large number of poles is required to produce the desired frequency.

� Steam turbines and gas turbines operate best at relatively high speeds, and turbine- driven alternators or turbine generators are commonly two- or four-pole cylindrical- rotor machines.

4

Figure 4.8 Elementary two-pole cylindrical-rotor field winding.

� Most of the world’s power systems are three-phase systems. With very few exceptions,

synchronous generators are three-phase machines. � A simplified schematic view of a three-phase, two-pole machine with one coil per phase

is shown in Fig. 4.12(a) � Fig. 4.12(b) depicts a simplified three-phase, four-pole machine. Note that a minimum

of two sets of coils must be used. In an elementary multipole machine, the minimum number of coils sets is given by one half the number of poles.

� Note that coils ),( aa − and ),( aa ′−′ can be connected in series or in parallel. Then the coils of the three phases may then be either Y- or �-connected. See Fig. 4.12(c).

Errore. Figure 4.12 Schematic views of three-phase generators: (a) two-pole, (b) four-pole, and

(c) Y connection of the windings.

� The electromechanical torque is the mechanism through which a synchronous generator converts mechanical to electric energy. � When a synchronous generator supplies electric power to a load, the armature current

creates a magnetic flux wave in the air gap that rotates at synchronous speed. � This flux reacts with the flux created by the field current, and an electromechanical

torque results from the tendency of these two magnetic fields to align. � In a generator this torque opposes rotation, and mechanical torque must be applied from

the prime mover to sustain rotation. � The counterpart of the synchronous generator is the synchronous motor.

� Ac current supplied to the armature winding on the stator, and dc excitation is supplied to the field winding on the rotor. The magnetic field produced by the armature currents rotates at synchronous speed. (Why?)

� To produce a steady electromechanical torque, the magnetic fields of the stator and rotor must be constant in amplitude and stationary with respect to each other.

� In a motor the electromechanical torque is in the direction of rotation and balances the opposing torque required to drive the mechanical load.

� In both generators and motors, an electromechanical torque and a rotational voltage are

5

produced which are the essential phenomena for electromechanical energy conversion. � Note that the flux produced by currents in the armature of a synchronous motor rotates

ahead of that produced by the field, thus pulling on the field (and hence on the rotor) and doing work. This is the opposite of the situation in a synchronous generator, where the field does work as its flux pulls on that of the armature, which is lagging behind.

� Induction Machines

� Alternating currents are applied directly to the stator windings. Rotors currents are then produced by induction, i.e., transformer action. � Alternating currents flow in the rotor windings of an induction machine, in contrast to a

synchronous machine in which a field winding on the rotor is excited with dc current. � The induction machine may be regarded as a generalized transformer in which electric

power is transformed between rotor and stator together with a change of frequency and a flow of mechanical power.

� The induction motor is the most common of all motors. � The induction machine is seldom used as a generator. � In recent years it has been found to be well suited for wind-power applications. � It may also be used as a frequency changer.

� In the induction motor, the stator windings are essentially the same as those of a synchronous machine. The rotor windings are electrically short-circuited. � The rotor windings frequently have no external connections. � Currents are induced by transformer action from the stator winding. � Squirrel-cage induction motor: relatively expensive and highly reliable.

� The armature flux in the induction motor leads that of the rotor and produces an electromechanical torque. � The rotor does not rotate synchronously. � It is the slipping of the rotor with respect to the synchronous armature flux that gives

rise to the induced rotor currents and hence the torque. � Induction motors operate at speeds less than the synchronous mechanical speed. � A typical speed-torque characteristic for an induction motor is shown in Fig. 4.15.

Figure 4.15 Typical induction-motor speed-torque characteristic. §4.2.2 DC Machines � DC Machines

� There are two sets of windings in a dc machine. � The armature winding is on the rotor with current conducted from it by means of carbon

brushes. � The field winding is on the stator and is excited by direct current.

� An elementary two-pole dc generator is shown in Fig. 4.17.

6

� Armature winding: ),( aa − , ο180pitch = � The rotor is normally turned at a constant speed by a source of mechanical power

connected the shaft.

Figure 4.17 Elementary dc machine with commutator.

� The air-gap flux distribution usually approximates a flat-topped wave, rather than the

sine wave found in ac machines, and is shown in Fig. 4.18(a). � Rotation of the coil generates a coil voltage which is a time function having the same

waveform as the spatial flux-density distribution. � The voltage induced in an individual armature coil is an alternating voltage and

rectification is produced mechanically by means of a commutator. Stationary carbon brushes held against the commutator surface connect the winding to the external armature terminal.

� The need for commutation is the reason why the armature windings are placed on the rotor.

� The commutator provides full-wave rectification, and the voltage waveform between brushes is shown in Fig. 4.18(b).

Figure 4.18 (a) Space distribution of air-gap flux density in an elementary dc machine;

(b) waveform of voltage between brushes.

� It is the interaction of the two flux distributions created by the direct currents in the field and the armature windings that creates an electromechanical torque.

7

� If the machine is acting as a generator, the torque opposes rotation. � If the machine is acting as a motor, the torque acts in the direction of the rotation.

§4.3 MMF of Distributed Windings � Most armatures have distributed windings, i.e. windings which are spread over a number of

slots around the air-gap periphery. � The individual coils are interconnected so that the result is a magnetic field having the

same number of poles as the field winding. � Consider Fig. 4.19(a).

� Full-pitch coil: a coil which spans 180 electrical degrees. � In Fig. 4.19(b), the air gap and winding are in developed form (laid out flat) and the

air-gap mmf distribution is shown by the steplike distribution of amplitude 2/Ni .

Figure 4.19 (a) Schematic view of flux produced by a concentrated, full-pitch winding in a machine

with a uniform air gap. (b) The air-gap mmf produced by current in this winding. §4.3.1 AC Machines � It is appropriate to focus our attention on the space-fundamental sinusoidal component of the

air-gap mmf. � In the design of ac machines, serious efforts are made to distribute the coils making up the

windings so as to minimize the higher-order harmonic components. � The rectangular air-gap mmf wave of the concentrated two-pole, full-pitch coil of Fig.

4.19(b) can be resolved to a Fourier series comprising a fundamental component and a series of odd harmonics. � The fundamental component aglF and its amplitude ag1 peak( )F are

agl

4cos

2 a

NiF θ

π� �= � �� �

(4.3)

8

( ) ��

���

�=2

4peakagl

NiF

π (4.4)

� Consider a distributed winding, consisting of coils distributed in several slots. � Fig. 4.20(a) shows phase a of the armature winding of a simplified two-pole,

three-phase ac machine and phases b and c occupy the empty slots. � The windings of the three phases are identical and are located with their magnetic

axes 120 degrees apart. The winding is arranged in two layers, each full-pitch coil of cN turns having one side in the top of a slot and the other coil side in the bottom of a slot a pole pitch away.

� Fig. 4.20(b) shows that the mmf wave is a series of steps each of height c a2N i . It can be seen that the distributed winding produces a closer approximation to a sinusoidal mmf wave than the concentrated coil of Fig. 4.19 does.

Figure 4.20 The mmf of one phase of a distributed two-pole, three-phase winding with full-pitch coils.

� The modified form of (4.3) for a distributed multipole winding is

w phagl a a

4 polescos

poles 2

k NF i θ

π� � � �= � � � �

� �� � (4.5)

phN : number of series turns per phase,

wk : winding factor, a reduction factor taking into account the distribution of the winding,

typically in the range of 0.85 to 0.95, w b p d p(or )k k k k k= . � The peak amplitude of this mmf wave is

9

w phag1 peak a

4( )

poles

k NF i

π� �

= � �� �

(4.6)

� Eq. (4.5) describes the space-fundamental component of the mmf wave produced by

current in phase a of a distributed winding. � If a m cosi I tω= the result will be an mmf wave which is stationary in space and

varies sinusoidally both with respect to aθ and in time. � The application of three-phase currents will produce a rotating mmf wave.

� Rotor windings are often distributed in slots to reduce the effects of space harmonics. � Fig. 4.21(a) shows the rotor of a typical two-pole round-rotor generator. � As shown in Fig. 4.21(b), there are fewer turns in the slots nearest the pole face. � The fundamental air-gap mmf wave of a multipole rotor winding is

r ragl r r

4 polescos

poles 2k N

F I θπ� � � �= � � � �

� �� � (4.7)

r rag1 peak r

4( )

polesk N

F Iπ� �

= � �� �

(4.8)

Figure 4.21 The air-gap mmf of a distributed winding on the rotor of a round-rotor generator.

10

§4.3.2 DC Machines � Because of the restrictions imposed on the winding arrangement by the commutator, the mmf

wave of a dc machine armature approximates a sawtooth waveform more nearly than the sine wave of ac machines. � Fig. 4.22 shows diagrammatically in cross section the armature of a two-pole dc machine.

� The armature coil connections are such that the armature winding produces a magnetic field whose axis is vertical and thus is perpendicular to the axis of the field winding.

� As the armature rotates, the magnetic field of the armature remains vertical due to commutator action and a continuous unidirectional torque results.

� The mmf wave is illustrated and analyzed in Fig. 4.23.

11

Figure 4.22 Cross section of a two-pole dc machine.

Figure 4.23 (a) Developed sketch of the dc machine of Fig. 4.22; (b) mmf wave; (c) equivalent

sawtooth mmf wave, its fundamental component, and equivalent rectangular current sheet.

� DC machines often have a magnetic structure with more than two poles. � Fig. 4.24(a) shows schematically a four-pole dc machine. � The machine is shown in laid-out form in Fig. 4.24(b).

12

Figure 4.24 (a) Cross section of a four-pole dc machine; (b) development of current sheet and mmf wave.

� The peak value of the sawtooth armature mmf wave can be written as

( ) aag apeak

A turns/pole2 poles

CF i

m� �

= ⋅� �⋅� � (4.9)

aC = total number of conductors in armature winding m = number of parallel paths through armature winding

ai = armature current, A

( ) aag a a apeak

, = /(2 ): no. of series armature turns poles

NF i N C m

� �= � �� �

(4.10)

( ) aag a2peak

8

polesN

F iπ

� �= � �

� � (4.11)

(4.12)

§4.4 Magnetic Fields In Rotating Machinery � The behavior of electric machinery is determined by the magnetic fields created by currents in

the various windings of the machine. � The investigations of both ac and dc machines are based on the assumption of sinusoidal

spatial distribution of mmf. � Results from examining a two-pole machine can immediately be extrapolated to a

multipole machine. §4.4.1 Magnetic with Uniform Air Gaps � Consider machines with uniform air gaps.

� Fig. 4.25(a) shows a single full-pitch, N-turn coil in a high-permeability magnetic structure ( )µ → ∞ , with a concentric, cylindrical rotor. � In Fig. 4.25(b) the air-gap mmf agF is plotted versus angle aθ .

� Fig. 4.25(c) demonstrates the air-gap constant radial magnetic field agH .

agag

FH

g= (4.12)

( ) ag1agl a

4cos

2

F NiH

g gθ

π� �

= = � �� �

(4.13)

13

( )agl peak

42Ni

Hgπ

� �= � �

� � (4.14)

� For a distributed winding such as that of Fig. 4.20, the air-gap magnetic field intensity is

w phagl a a

4 polescos

poles 2

k NH i

gθ

π� � � �= � � � �⋅ � �� �

(4.15)

14

Figure 4.25 The air-gap mmf and radial component of Hag for a concentrated full-pitch winding.

§4.4.2 Machines with Nonuniform Air Gaps � The air-gap magnetic-field distribution of machines with nonuniform air gaps is more complex

than that of uniform-air-gap machines. � Fig. 4.26(a) shows the structure of a typical dc machine and Fig. 4.26(b) shows the

structure of a typical salient-pole synchronous machine.

Figure 4.26 Structure of typical salient-pole machines: (a) dc machine and (b) salient-pole synchronous machine.

� Detailed analysis of the magnetic field distributions requires complete solutions of the field problem. � Fig. 4.27 shows the magnetic field distribution in a salient-pole dc generator

(obtained by finite-element solution).

15

Figure 4.27 Finite-element solution of the magnetic field distribution in a salient-pole dc generator.

Field coils excited; no current in armature coils. (General Electric Company.)

§4.5 Rotating MMF Waves in AC Machines � To understand the theory and operation of polyphase ac machines, it is necessary to study the

nature of the mmf wave produced by a polyphase winding. §4.5.1 MMF Wave of a Single-Phase Winding � Fig. 4.28(a) shows the space-fundamental mmf distribution of a single-phase winding.

� Note that from Eq. (4.5), ag1F is

w phagl a a

4 polescos

poles 2

k NF i θ

π� � � �= � � � �

� �� � (4.16)

When the winding is exicted by a current

a a ecosi I tω= (4.17) the mmf distribution is given by

( )

agl max a e

max ae e

polescos cos

2

cos cos

F F t

F t

θ ω

θ ω

� �= � �� �

= (4.18)

w phmax a

4poles

k NF I

π� �

= � �� �

(4.19)

� This mmf distribution remains fixed in space with an amplitude that varies sinusoidally in time at frequency eω , as shown in Fig. 4.28(a).

� The air-gap mmf of a single-phase winding exicted by a source of ac current can be resolved into rotating traveling waves. � By the identity 1 1

2 2cos cos cos( ) cos( )α β α β α β= − + + ,

( ) ( )agl max ae e ae e

1 1cos cos

2 2F F t tθ ω θ ω� �= − + + � �

(4.20)

( )ag1 max ae e

1cos

2F F tθ ω+ = − (4.21)

( )ag1 max ae e

1cos

2F F tθ ω− = + (4.22)

16

� ag1F + travels in the aθ+ direction and ag1F − travels in the aθ− direction. � This decomposition is shown graphically in Fig. 4.28(b) and in a phasor

representation in Fig. 4.28(c).

Figure 4.28 Single-phase-winding space-fundamental air-gap mmf: (a) mmf distribution of a

single-phase winding at various times; (b) total mmf aglF decomposed into two traveling waves F −

and F + ; (c) phasor decomposition of aglF . §4.5.2 MMF Wave of a Polyphase Winding � We are to study the mmf distribution of three-phase windings such as those found on the stator

of three-phase induction and synchronous machines. � In a three-phase machine, the windings of the individual phases are displaced from each

other by 120 electrical degrees in space around the air-gap circumference as shown in Fig. 4.29 in which the concentrated full-pitch coils may be considered to represent distributed windings. � Under balanced three-phase conditions, the excitation currents (Fig. 4.30) are

a m ecosi I tω= (4.23)

( )b m ecos 120i I tω= − o (4.24)

( )c m ecos 120i I tω= + o (4.25)

17

Figure 4.29 Simplified two-pole three-phase stator winding.

Figure 4.30 Instantaneous phase currents under balanced three-phase conditions.

� The mmf of phase a has been shown to be

a1 a1 a1F F F+ −= + (4.26)

( )+a1 max ae e

1cos

2F F tθ ω= − (4.27)

( )a1 max ae e

1cos

2F F tθ ω− = + (4.28)

w phmax m

4poles

k NF I

π� �

= � �� �

(4.29)

� Similarly, for phases b and c

b1 b1 b1F F F+ −= + (4.30)

( )+b1 max ae e

1cos

2F F tθ ω= − (4.31)

18

( )b1 max ae e

1cos 120

2F F tθ ω− = + + o (4.32)

c1 c1 c1F F F+ −= + (4.33)

( )+c1 max ae e

1cos

2F F tθ ω= − (4.34)

( )c1 max ae e

1cos 120

2F F tθ ω− = + − o (4.35)

� The total mmf is the sum

( )ae a1 b1 c1,F t F F Fθ = + + (4.36) It can be performed in terms of the positive- and negative- traveling waves.

( )

( ) ( ) ( )ae a1 b1 c1

max ae e ae e ae e

,

1cos cos 120 cos 120

20

F t F F F

F t t t

θ

θ ω θ ω θ ω

− − − −= + +

� �= + + + − + + + ��

=

o o (4.37)

( )ae a1 b1 c1

max ae e

,

3cos( )

2

F t F F F

F t

θ

θ ω

+ + + += + +

= − (4.38)

� The result of displacing the three windings by 120o in space phase and displacing the winding currents by 120o in time phase is a single positive-traveling mmf wave

( ) ( )ae max ae e

max a e

3, cos

23 poles

cos2 2

F t F t

F t

θ θ ω

θ ω

= −

� �� �= −� �� �� �� �

(4.39)

� Under balanced three-phase conditions, the three-phase winding produces an air-gap mmf wave which rotates at synchronous angular velocity sω (rad/sec)

s e

2poles

ω ω� �= � �� �

(4.40)

eω : angular velocity of the applied electrical excitation (rad/sec) � sn : synchronous speed

e e /(2 )f ω π= : applied electrical frequency

r/min poles120

es fn ���

����

�= (4.41)

� A polyphase winding exicted by balanced polyphase currents produces a rotating mmf

wave. � It is the interaction of this magnetic flux wave with that of the rotor which produces

torque. � Constant torque is produced when rotor-produced magnetic flux rotates in

synchronism with that of the stator. §4.5.3 Graphical Analysis of Polyphase MMF � For balanced three-phase currents, the production of a rotating mmf can also be shown

graphically.

19

� Refer to Fig. 4.30 and Fig. 4.31. � As time passes, the resultant mmf wave retains its sinusoidal form and amplitude but

rotates progressively around the air gap. � The net result is an mmf wave of constant amplitude rotating at uniform angular

velocity.

Figure 4.31 The production of a rotating magnetic field by means of three-phase currents.

� Practice Problem 4.3

Repeat Example 4.3 for a three-phase stator excited by balanced 50-Hz currents.

§4.6 Generated Voltage

§4.7 Torque in Nonsalient-Pole Machines

20

§4.6 Generated Voltage

§4.6.1 AC Machines

Figure 4.32 Cross-sectional view of an elementary three-phase ac machine.

fff I

Nk

gB ��

�

����

�=

poles4 0

peak πµ

(4.42)

��

���

�= rBB θ2

polescospeak (4.43)

lrB

rdBl rr

peak

poles/

poles/ peak

2poles

2

2poles

cos

���

����

�=

��

���

�=Φ +

−θθ

π

π

(4.44)

tNk

tNk

mepphw

mpphwa

ω

ωλ

cos

2poles

cos

Φ=

���

����

���

���

�Φ= (4.45)

mme ωω ��

���

�=2

poles (4.46)

tNktdt

dNk

dtd

e mepphwmemep

phwa

a ωωωλsincos Φ−

Φ== (4.47)

tNke mepphwmea ωω sinΦ−= (4.48)

21

pphwmepphwme NkfNkE Φ=Φ= πω 2max (4.49)

pphwmepphwmerms NkfNkfE Φ=Φ= 22

2π (4.50)

22

( ) ( ) pmememepmea NtdtNE Φ=Φ= ωπ

ωωωπ

π 2sin

10

(4.51)

��

���

�Φ=Φ��

���

�=30

polespoles n

NNE pmpa ωπ

(4.52)

nmC

mC

E pa

mpa

a ��

���

���

���

�=��

���

���

���

�=60

poles2

poles ωπ

(4.53)

§4.7 Torque in Nonsalient-pole Machines

1

Chapter 5 Synchronous Machines � Main features of synchronous machines:

� A synchronous machine is an ac machine whose speed under steady-state conditions is proportional to the frequency of the current in its armature.

� The rotor, along with the magnetic field created by the dc field current on the rotor, rotates at the same speed as, or in synchronism with, the rotating magnetic field produced by the armature currents, and a steady torque results.

Errore. Figure 4.12 Schematic views of three-phase generators: (a) two-pole, (b) four-pole, and

(c) Y connection of the windings. §5.1 Introduction to Polyphase Synchronous Machines � Synchronous machines:

� Armature winding: on the stator, alternating current. � Field winding: on the rotor, dc power supplied by the excitation system.

� Cylindrical rotor: for two- and four-pole turbine generators. � Salient-pole rotor: for multipolar, slow-speed, hydroelectric generators and for most

synchronous motors. � Acting as a voltage source:

� Frequency determined by the speed of its mechanical drive (or prime mover). � The amplitude of the generated voltage is proportional to the frequency and the field

current.

tNk

tNk

mepphw

mpphwa

ω

ωλ

cos

2poles

cos

Φ=

���

����

���

���

�Φ= (4.45)

2

mme 2poles ωω �

�

���

�= (4.46)

tNktdt

dNk

dtd

e mepphwmemep

phwa

a ωωωλsincos Φ−

Φ== (4.47)

tNke mepphwmea ωω sinΦ−= (4.48)

pphwmepphwmemax 2 Φ=Φ= NkfNkE πω (4.49)

pphwmepphwmerms 22

2 Φ=Φ= NkfNkfE ππ (4.50)

� Synchronous generators can be readily operated in parallel: interconnected power

systems. � When a synchronous generator is connected to a large interconnected system containing

many other synchronous generators, the voltage and frequency at its armature terminals are substantially fixed by the system. � It is often useful, when studying the behavior of an individual generator or group of

generators, to represent the remainder of the system as a constant-frequency, constant-voltage source, commonly referred to as an infinite bus.

� Analysis of a synchronous machine connected to an infinite bus. � Torque equation:

RFfR

2

sin2

poles2

δπFT Φ�

�

���

�−= (5.1)

where RΦ = resultant air-gap flux per pole

fF = mmf of the dc field winding

RFδ = electric phase angle between magnetic axes of RΦ and fF

� The minus sign indicates that the electromechanical torque acts in the direction to bring the interacting fields into alignment.

� In a generator, the prime-mover torque acts in the direction of rotation of the rotor, and the electromechanical torque opposes rotation. The rotor mmf wave leads the resultant air-gap flux.

� In a motor, the electromechanical torque is in the direction of rotation, in opposition to the retarding torque of the mechanical load on the shaft.

� Torque-angle curve: Fig. 5.1.

Figure 5.1 Torque-angle characteristics.

3

� An increase in prime-mover torque will result in a corresponding increase in the torque angle.

� maxTT = : pull-out torque at ο90RF =δ . Any further increase in prime-mover torque cannot be balanced by a corresponding increase in synchronous electromechanical torque, with the result that synchronism will no longer be maintained and the rotor will speed up. � loss of synchronism, pulling out of step.

§5.2 Synchronous-Machine Inductances; Equivalent Circuits

Figure 5.2 Schematic diagram of a two-pole,

three-phase cylindrical-rotor synchronous machine. §5.2.1 Rotor Self-Inductance

§5.2.2 Stator-to-Rotor Mutual Inductances

§5.2.3 Stator Inductances; Synchronous Inductance §5.2.4 Equivalent Circuit � Equivalent circuit for the synchronous machine:

� Single-phase, line-to-neutral equivalent circuits for a three-phase machine operating under balanced, three-phase conditions.

sL = effective inductance seen by phase a under steady-state, balanced three-phase machine operating conditions.

ses LX ω= : synchronous reactance

aR = armature winding resistance

afe = voltage induced by the field winding flux (generated voltage, internal voltage)

aI = armature current

av = terminal voltage Motor reference direction:

faasaaaˆˆˆˆ EIjXIRV ++= (5.23)

Generator reference direction:

faasaaaˆˆˆˆ EIjXIRV +−−= (5.24)

4

Figure 5.3 Synchronous-machine equivalent circuits:

(a) motor reference direction and (b) generator reference direction.

ϕXXX += als (5.25)

alX = armature leakage reactance

ϕX = magnetizing reactance of the armature winding

RE = air-gap voltage or the voltage behind leakage reactance

Figure 5.4 Synchronous-machine equivalent circuit showing air-gap and

leakage components of synchronous reactance and air-gap voltage.

5

§5.4 Steady-State Power-Angle Characteristics � The maximum power a synchronous machine can deliver is determined by the maximum

torque that can be applied without loss of synchronism with the external system to which it is connected. � Both the external system and the machine itself can be represented as an impedance in

series with a voltage source.

Figure 5.11 (a) Impedance interconnecting two voltages; (b) phasor diagram.

6

φcos22 IEP = (5.34)

ZEE

I 21ˆˆ

ˆ −= (5.35)

δjeEE 11ˆ = (5.36)

22ˆ EE = (5.37)

ZjeZXjRZ φ=+= (5.38)

( ) ZZ

Z

jjj

jj e

ZE

eZE

eZEeE

IeI φφδφ

δφ −− −=−== 2121ˆ (5.39)

( ) ( )ZZ ZE

ZE

I φφδφ −−−= coscoscos 21 (5.40)

( )2

2221

2 cosZ

REZEE

P Z −−= φδ (5.41)

( )2

2221

2 sinZ

REZEE

P Z −+= αδ (5.42)

where

��

���

�=−= −

XR

ZZ1tan90 φα ο (5.43)

( )2

2121

1 sinZ

REZEE

P Z −−= φδ (5.44)

Frequently, 0and, ≈≈<< ZXZZR α ,

δsin2121 X

EEPP == (5.45)

� Equation (5.45) is commonly referred to as the power-angle characteristic for a

synchronous machine. � The angle δ is known as the power angle. � Note that 1E and 2E are the line-to-neutral voltages. � For three-phase systems, a factor “3” shall be placed in front of the equation. � The maximum power transfer is

XEE

PP 21max,2max,1 == (5.46)

occurring when ο90±=δ . � If 0>δ , 1E leads 2E and power flows from source 1E to 2E .

� When 0<δ , 1E lags 2E and power flows from source 2E to 1E .

� Consider Fig. 5.12 in which a synchronous machine with generated voltage afE

and synchronous sX is connected to a system whose Thevenin equivalent is a

voltage source EQV in series with a reactive impedance EQjX . The power-angle characteristic can be written

δsinEQs

EQfa

XX

VEP

+= (5.47)

7

Figure 5.12 Equivalent-circuit representation of

a synchronous machine connected to an external system.

� Note that 21EEP ∝ , 1−∝ XP , 21max EEP ∝ , and 1max

−∝ XP .

� In general, stability considerations dictate that a synchronous machine achieve steady-state operation for a power angle considerably less than ο90 .

8

Figure 5.14 Equivalent circuits and phasor diagrams for Example 5.7.

9

10

11

§5.3 Open- and Short-Circuit Characteristics §5.3.1 Open-Circuit Saturation Characteristic and No-Load Rotational Losses

Figure 5.5 Open-circuit characteristic of a synchronous machine.

§5.3.2 Short-Circuit Characteristic and Load Loss

12

Figure 5.6 Typical form of an open-circuit core-loss curve.

( )saafa jXRIE += ˆˆ (5.26)

Figure 5.7 Open- and short-circuit characteristics of a synchronous machine.

Figure 5.8 Phasor diagram for short-circuit conditions.

( )laaaR jXRIE += ˆˆ (5.27)

aca

agaus I

VX

,

,, = (5.28)

a

ratedas I

VX

′= , (5.29)

13

Figure 5.9 Open- and short-circuit characteristics showing

equivalent magnetization line for saturated operating conditions.

fOfO

′′′

=SCR (5.30)

AFSCAFNL

SCR = (5.31)

14

Figure 5.10 Typical form of short-circuit load loss and stray load-loss curves.

tT

RR

t

T

++=

5.2345.234

(5.32)

( )2eff,ntaturecurrecircuitarmshort

loss loadcircuit short−

−=aR (5.33)

15

§5.5 Steady-State Operating Characteristics

Figure 5.15 Characteristic form of synchronous-generator compounding curves.

16

Figure 5.16 Capability curves of an 0.85 power factor, 0.80 short-circuit ratio, hydrogen-cooled turbine generator. Base MVA is rated MVA at 0.5 psig hydrogen.

aa IVQP =+= 22powerApparent (5.48)

Figure 5.17 Construction used for the derivation of a synchronous generator capability curve.

asa IjXVQjP ˆˆ +=− (5.49)

asafa IjXVE ˆˆˆ += (5.50) 222

2���

����

�=��

�

����

�++

s

faa

s

a

X

EV

XV

QP (5.51)

Figure 5.18 Typical form of synchronous-generator V curves.

17

Figure 5.19 Losses in a three-phase, 45-kVA, Y-connected,

220-V, 60-Hz, six-pole synchronous machine.

18

§5.6 Effects of Salient Poles; Introduction to Direct-And Quadrature-Axis Theory §5.6.1 Flux and MMF Waves

19

Figure 5.20 Direct-axis air-gap fluxes in a salient-pole synchronous machine.

( )33,3 3cos2 φω += tVE ea (5.52)

( )( ) ( )3333,3 3cos21203cos2 φωφω +=+−= tVtVE eebο (5.53)

( )( ) ( )3333,3 3cos21203cos2 φωφω +=+−= tVtVE eecο (5.54)

Figure 5.21 Quadrature-axis air-gap fluxes in a salient-pole synchronous machine.

Figure 5.22 Phasor diagram of a salient-pole synchronous generator.

§5.3.2 Phasor Diagrams for Salient-Pole Machines

20

Figure 5.23 Phasor diagram for a synchronous generator showing

the relationship between the voltages and the currents.

dlad XXX ϕ+= (5.55)

qlaq XXX ϕ+= (5.56)

Figure 5.24 Relationships between component voltages in a phasor diagram.

baab

oaao ′′

=′′

(5.57)

aqa

q

qqIXI

I

XIˆˆ

ˆ

ˆoa

baab

ao ==��

���

� ′′=′′ (5.58)

qqddaaafa IjXIjXIRVE ˆˆˆˆˆ +++= (5.59)

21

Figure 5.25 Generator phasor diagram for Example 5.9.

22

1

Chapter 6 Polyphase Induction Machines � Study on the behavior of polyphase induction machines:

� The analysis begins with the development of single-phase equivalent circuits. � The general form is suggested by the similarity of an induction machine to a

transformer. � The equivalent circuits can be used to study the electromechanical characteristics of

an induction machine as well as the loading presented by the machine on its supply source.

§6.1 Introduction to Polyphase Induction Machines � An induction machine is one in which alternating current is supplied to the stator directly and

to the rotor by induction or transformer action from the stator. � The stator winding is excited from a balanced polyphase source and produces a magnetic

field in the air gap rotating at synchronous speed. � The rotor winding may one of two types.

� A wound rotor is built with a polyphase winding similar to, and wound with the same number of poles as, the stator. The rotor terminals are available external to the motor.

� A squirrel-cage rotor has a winding consisting of conductor bars embedded in slots in the rotor iron and short-circuited at each end buy conducting end rings. It is the most commonly used type of motor in sizes ranging from fractional horsepower on up.

� The difference between synchronous speed and the rotor speed is commonly referred to as the slip of the rotor. The fractional slip s is

s

s

n ns

n−= (6.1)

� The slip is often expressed in percent. � n : rotor speed in rpm ( ) snsn −= 1 (6.2) � mω : mechanical angular velocity

( ) sm s ωω −= 1 (6.3) � rf : the frequency of induced voltages, the slip frequency

r ef s f= (6.4) – A wound-rotor induction machine can be used as a frequency changer.

� The rotor currents produce an air-gap flux wave that rotates at synchronous speed and in synchronism with that produced by the stator currents. � With the rotor revolving in the same direction of rotation as the stator field, the rotor

currents produce a rotating flux wave rotating at ssn with respect to the rotor in the forward direction.

� With respect to the stator, the speed of the flux wave produced by the rotor currents (with frequency esf ) equals

( )s s s s1sn n sn n s n+ = + − = (6.5) � Because the stator and rotor fields each rotate synchronously, they are stationary with

respect to each other and produce a steady torque, thus maintaining rotation of the rotor. Such torque is called an asynchronous torque.

2

� Equation (4.81) 2

sr r r

polessin

2 2T F

π δ� �= − Φ� �� �

can be expressed in the form

r rsinT KI δ= − (6.6) rI : the rotor current

rδ : the angle by which the rotor mmf wave leads the resultant air-gap mmf wave � Fig. 6.4 shows a typical polyphase squirrel-cage induction motor torque-speed curve.

The factors influencing the shape of this curve can be appreciated in terms of the torque equation.

Figure 6.4 Typical induction-motor torque-speed

curve for constant-voltage, constant-frequency operation.

� Under normal running conditions the slip is small: 2 to 10 percent at full load. � The maximum torque is referred to as the breakdown torque. � The slip at which the peak torque occurs is proportional to the rotor resistance.

§6.2 Currents and Fluxes in Polyphase Induction Machines

§6.3 Induction-Motor Equivalent Circuit � Only machines with symmetric polyphase windings exited by balanced polyphase voltages are

considered. It is helpful to think of three-phase machines as being Y-connected.

� Stator equivalent circuit: ( )11121

ˆˆˆ jXRIEV ++= (6.8)

1

2

1

1

1

ˆ Stator line-to-neutral terminal voltageˆ Counter emf (line-to-neutral) generated by the resultant air-gap fluxˆ Stator current

Stator effective resistance

Stator leakage reactance

V

E

I

R

X

=

=

===

3

Figure 6.7 Stator equivalent circuit for a polyphase induction motor.

� Rotor equivalent circuit:

2

22 ˆ

ˆ

I

EZ = (6.9)

2 22s rotor2s eff eff rotor

2s rotor

ˆ ˆˆ ˆE E

Z N N ZI I

� �= = =� �

� � (6.10)

2sZ : the slip-frequency leakage impedance of the equivalent rotor rotorZ : the slip-frequency leakage impedance

2s2s 2 2

2s

ˆˆE

Z R jsXI

= = + (6.11)

2R = Referred rotor resistance 2sR = Referred rotor leakage reactance at slip frequency

2X = Referred rotor leakage reactance at stator frequency ef

Figure 6.8 Rotor equivalent circuit for a polyphase induction motor at slip frequency.

22ˆˆ II s = (6.12)

22 sEE s = (6.13)

22ˆˆ EsE s = (6.14)

2222

2

2

2

ˆ

ˆ

ˆ

ˆjsXRZ

I

Es

I

Es

s

s +=== (6.15)

22

2

22 ˆ

ˆjX

sR

I

EZ +== (6.16)

4

� Fig. 6.9 shows the single-phase equivalent circuit.

Figure 6.9 Single-phase equivalent circuit for a polyphase induction motor.

§6.4 Analysis of the Equivalent Circuit � The single-phase equivalent circuit can be used to determine a wide variety of steady-state

performance characteristics of polyphase induction machines. � gapP : the total power transferred across the air gap from the stator

rotorP : the total rotor ohmic loss

��

���

�=s

RInP 22

2phgap (6.17)

222phrotor RInP s= (6.18)

222phrotor RInP = (6.19)

222ph

222phrotorgapmech RIn

sR

InPPP −��

���

�=−= (6.20)

��

���

� −=s

sRInP

12

22phmech (6.21)

( ) gapmech 1 PsP −= (6.22)

rotor gapP sP= (6.23)

� Of the total power delivered across the air gap to the rotor, the fraction 1 s− is converted to mechanical power and the fraction s is dissipated as ohmic loss in the rotor conductors.

� When power aspects are to be emphasized, the equivalent circuit can be redrawn in the manner of Fig. 6.10.

Figure 6.10 Alternative form of equivalent circuit.

5

� Consider the electromechanical torque mechT . ( ) mechmechmech 1 TsTP sm ωω −== (6.24)

( )s

sRInPPT

ωωω/2

22ph

s

gap

m

mechmech === (6.25)

ee

s

f ωπω ���

����

�==

poles2

poles4

(6.26)

rotmechshaft PPP −= (6.27)

rotmechm

shaftshaft TT

PT −==

ω (6.28)

Figure 6.11 Equivalent circuits with the core-loss resistance Rc neglected corresponding to

(a) Fig. 6.9 and (b) Fig. 6.10.

6

7

§6.5 Torque and Power by Use of Thevenin’s Theorem � Considerable simplification will be obtained from application of Thevenin’s network theorem

to the induction-motor equivalent circuit.

Figure 6.12 (a) General linear network and

(b) its equivalent at terminals ab by Thevenin’s theorem.

Figure 6.13 Induction-motor equivalent circuits simplified by Thevenin’s theorem.

( )����

���

�

++=

m

m

XXjRjX

VV11

1eq1,ˆˆ (6.29)

( )1,eq 1,eq 1,eq 1 1 in parallel with mZ R jX R jX jX= + = + (6.30)

( )( )m

m

XXjRjXRXj

VZ++

+=

11

111eq1,ˆ (6.31)

sRjXZ

VI

/

ˆˆ

22eq1,

eq,12 ++

= (6.32)

( )( )( ) ( ) �

��

�

�

+++= 2

2eq1,2

2eq1,

22eq1,ph

mech/

/1

XXsRR

sRVnT

sω (6.33)

� The general shape of the torque-speed or torque-slip curve with motor connected to a

constant-voltage, constant-frequency source is shown in Figs. 6.14 and 6.15.

8

Figure 6.14 Induction-machine torque-slip curve showing braking, motor, and generator regions.

Figure 6.15 Computed torque, power, and current curves for the 7.5-kW motor in Exps 6.2 and 6.3.

� Maximum electromechanical torque will occur at a value of slip maxTs for which

( )2221,eq 1,eq 2

maxT

RR X X

s= + + (6.34)

( )2

max T 221,eq 1,eq 2

Rs

R X X=

+ + (6.35)

( ) ��

�

�

�

+++=

22eq1,

2eq1,eq1,

2eq1,

max

5.01

XXRR

VnT ph

sω (6.36)

9

10

Figure 6.16 Induction-motor torque-slip curves showing effect of changing rotor-circuit resistance.

§6.5 Parameter Determination from No-Load and Blocked-Rotor Tests

� The equivalent-circuit parameters needed for computing the performance of a poly-phase

induction motor under load can be obtained from the results of a no-load test, a blocked-rotor test, and measurement of the dc resistances of the stator windings.

§6.6.1 No-Load Test

� Like the open-circuit test on a transformer, the no-load test on an induction motor gives information with respect to exciting current and no-load losses.

§6.6.2 Blocked-Rotor Test

� Like the short-circuit test on a transformer, the blocked-rotor test on an induction motor give information with respect to the leakage impedances.

11

12

1

Chapter 7 DC Machines � Dc machines are characterized by their versatility.

� By means of various combinations of shunt-, series-, and separately-excited field windings they can be designed to display a wide variety of volt-ampere or speed-torque characteristics for both dynamic and steady-state operation.

� Because of the ease with which they can be controlled, systems of dc machines have been frequently used in applications requiring a wide range of motor speeds or precise control of motor output.

§7.1 Introduction � The essential features of a dc machine are shown schematically in Fig. 7.1.

� Fig. 7.1(b) shows the circuit representation of the machine. � The stator has salient poles and is excited by one or more field coils.

� The air-gap flux distribution created by the field windings is symmetric about the center line of the field poles. This axis is called the field axis or direct axis.

� The ac voltage generated in each rotating armature coil is converted to dc in the external armature terminals by means of a rotating commutator and stationary brushes to which the armature leads are connected. � The commutator-brush combination forms a mechanical rectifier, resulting in a dc

armature voltage as well as an armature-mmf wave which is fixed in space. � The brushes are located so that commutation occurs when the coil sides are in the

neutral zone, midway between the field poles. � The axis of the armature-mmf wave is 90 electrical degrees from the axis of the field

poles, i.e., in the quadrature axis. � The armature-mmf wave is along the brush axis.

Figure 7.1 Schematic representations of a dc machine.

� Recall equation (4.81). Note that the torque is proportional to the product of the magnitudes of the interacting fields and to the sine of the electrical space angle between their magnetic axes. The negative sign indicates that the electromechanical torque acts in a direction to decrease the displacement angle between the fields.

2

sr r r

polessin

2 2T F

π δ� �= − Φ� �� �

(4.81)

2

� For the dc machine, the electromagnetic torque mechT can be expressed in terms of the interaction of the direct-axis air-gap per pole dΦ and the space-fundamental component a1F of the armature-mmf wave, in a form similar to (4.81). Note that

rsin 1δ = . 2

mech d a1

pole2 2

T Fπ � �= Φ� �� �

(7.1)

mech a d aT K i= Φ (7.2)

aa

poles 2

CK

�m= (7.3)

aK : a constant determined by the design of the winding, the winding constant

ai = current in external armature circuit

aC = total number of conductors in armature winding, m = number of parallel paths through winding

� The rectified voltage ae between brushes, known also as the speed voltage, is

a a d me K ω= Φ (7.4) � The generated voltage as observed from the brushes is the sum of the rectified

voltage of all the coils in series between brushes and is shown by the rippling line labeled ae in Fig. 7.2.

� With a dozen or so commutator segments per pole, the ripple becomes very small and the average generated voltage observed from the brushes equals the sum of the average values of the rectified coils voltages.

Figure 7.2 Rectified coil voltages and resultant voltage between brushes in a dc machine.

� Note that the electric power equals the mechanical power.

a a mech me i T ω= (7.5) � The flux-mmf characteristic is referred to as the magnetization curve.

� The direct-axis air-gap flux is produced by the combined mmf f fN i� of the field winding.

� The form of a typical magnetization curve is shown in Fig. 7.3(a). � The dashed straight line through the origin coinciding with the straight portion of the

magnetization curves is called the air-gap line. � It is assumed that the armature mmf has no effect on the direct-axis flux because the

axis of the armature-mmf wave is along the quadrature axis and hence perpendicular to the field axis. (This assumption needs reexamining!)

� Note the residual magnetism in the figure. The magnetic material of the field does not fully demagnetize when the net field mmf is reduced to zero.

� It is usually more convenient to express the magnetization curve in terms of the

3

armature emf a0e at a constant speed m0ω as shown in Fig. 7.3(b).

a a0a d

m m0

e eK

ω ω= Φ = (7.6)

ma a0

m0

( )e eωω

= (7.7)

a a00

( )n

e en

= (7.8)

� Fig. 7.3(c) shows the magnetization curve with only one field winding excited. This curve can easily be obtained by test methods.

Figure 7.3 Typical form of magnetization curves of a dc machine.

� Various methods of excitation of the field windings are shown in Fig. 7.4.

Figure 7.4 Field-circuit connections of dc machines:

(a) separate excitation, (b) series, (c) shunt, (d) compound.

� Consider first dc generators. � Separately-excited generators. � Self-excited generators: series generators, shunt generators, compound generators.

� With self-excited generators, residual magnetism must be present in the machine iron to get the self-excitation process started.

4

� N.B.: long- and short-shunt, cumulatively and differentially compound. � Typical steady-state volt-ampere characteristics are shown in Fig. 7.5, constant-speed

operation being assumed. � The relation between the steady-state generated emf aE and the armature terminal

voltage aV is

a a a aV E I R= − (7.10)

Figure 7.5 Volt-ampere characteristics of dc generators.

� Any of the methods of excitation used for generators can also be used for motors.

� Typical steady-state dc-motor speed-torque characteristics are shown in Fig. 7.6, in which it is assumed that the motor terminals are supplied from a constant-voltage source.

� In a motor the relation between the emf aE generated in the armature and and the armature terminal voltage aV is

a a a aV E I R= + (7.11)

a aa

a

V EI

R−= (7.12)

� The application advantages of dc machines lie in the variety of performance characteristics offered by the possibilities of shunt, series, and compound excitation.

Figure 7.6 Speed-torque characteristics of dc motors.

5

§7.4 Analytical Fundamentals: Electric-Circuit Aspects � Analysis of dc machines: electric-circuit and magnetic-circuit aspects

� Torque and power: The electromagnetic torque mechT

adamech IKT Φ= (7.13) The generated voltage aE

mdaa ωΦ= KE (7.14)

mC

Kπ2

poles aa = (7.15)

aa IE : electromagnetic power

adam

aamech IK

IET Φ==

ω (7.16)

Note that the electromagnetic power differs from the mechanical power at the machine shaft by the rotational losses and differs from the electric power at the machine terminals by the shunt-field and armature RI 2 losses.

� Voltage and current (Refer to Fig. 7.12.):

aV : the terminal voltage of the armature winding

tV : the terminal voltage of the dc machine, including the voltage drop across the series-connected field winding

ta VV = if there is no series field winding

aR : the resistance of armature, sR : the resistance of the series field

aaaa RIEV ±= (7.17) ( )saaat RRIEV +±= (7.18)

faL III ±= (7.19)

Figure 7.12 Motor or generator connection diagram with current directions.

6

7

� For compound machines, Fig. 7.12 shows a long-shunt connection and the short-shunt connection is illustrated in Fig. 7.13.

Figure 7.13 Short-shunt compound-generator connections.

§7.5 Analytical Fundamentals: Magnetic-Circuit Aspects � The net flux per pole is that resulting from the combined mmf’s of the field and armature

windings. � First we consider the mmf intentionally placed on the stator main poles to create the

working flux, i.e., the main-field mmf, and then we include armature-reaction effects. §7.5.1 Armature Reaction Neglected � With no load on the machine or with armature-reaction effects ignored, the resultant mmf is the

algebraic sum of the mmf’s acting on the main or direct axis.

f f s sMain field mmf N I N I− = ± (7.20)

sf s

f

Gross mmf equivalent shunt-field amperesN

I IN

� �= + � �

� � (7.21)

� An example of a no-load magnetization characteristic is given by the curve for 0=aI in

Fig. 7.14. � The generated voltage aE at any speed mω is given by

ma0

m0aE E

ωω� �

= � �� �

(7.22)

00

aa Enn

E ���

����

�= (7.23)

8

Figure 7.14 Magnetization curves for a 100-kW, 250-V, 1200-r/min dc machine.

Also shown are field-resistance lines for the discussion of self-excitation in § 7.6.1.

9

§7.5.2 Effects of Armature Reaction Included � Current in the armature winding gives rise to a demagnetizing effect caused by a cross-

magnetizing armature reaction. � One common approach is to base analyses on the measured performance of the machine.

� Data are taken with both the field and armature excited, and the tests are conducted so that the effects on generated emf of varying both the main-field excitation and armature mmf can be noted.

� Refer to Fig. 7.14. The inclusion of armature reaction becomes simply a matter of using the magnetization curve corresponding to the armature current involved.

� The load-saturation curves are displaced to the right of the no-load curve by an amount which is a function of aI .

� The effect of armature reaction is approximately the same as a demagnetizing mmf

arF acting on the main-field axis.

ar f f s sNet mmf gross mmf F N I N I AR= − = + − (7.24)

� Over the normal operating range, the demagnetizing effect of armature reaction may be assumed to be approximately proportional to the armature current.

� The amount of armature of armature reaction present in Fig. 7.14 is definitely more than one would expect to find in a normal, well-designed machine operating at normal currents.

10

§7.6 Analysis of Steady-State Performance � Generator operation and motor operation

� For a generator, the speed is usually fixed by the prime mover, and problems often encountered are to determine the terminal voltage corresponding to a specified load and excitation or to find the excitation required for a specified load and terminal voltage.

� For a motor, problems frequently encountered are to determine the speed corresponding to a specific load and excitation or to find the excitation required for specific load and speed conditions; terminal voltage is often fixed at the value of the available source.

§7.6.1 Generator Analysis � Analysis is based on the type of field connection.

� Separately-excited generators are the simplest to analyze. � Its main-field current is independent of the generator voltage. � For a given load, the equivalent main-field excitation is given by (7.21) and the

associated armature-generated voltage aE is determined by the appropriate magnetization curve.

� The voltage aE , together with (7.17) or (7.18), fixes the terminal voltage. � Shunt-excited generators will be found to self-excite under properly chosen operating

condition under which the generated voltage will build up spontaneously. � The process is typically initiated by the presence of a small amount of residual

magnetism in the field structure and the shunt-field excitation depends on the terminal voltage. Consider the field-resistance line, the line 0a in Fig. 7.14.

� The tendency of a shunt-connected generator to self-excite can be observed by examining the buildup of voltage for an unloaded shunt generator. – Buildup continues until the volt-ampere relations represented by the

magnetization curve and the field-resistance line are simultaneously satisfied.

Figure 7.15 Equivalent circuit for analysis of voltage buildup in a self-excited dc generator.

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Note that in Fig. 7.15,

( ) ( )fa f a a f f

diL L e R R i

dt+ = − + (7.25)

� The field resistance line should also include the armature resistance. � Notice that if the field resistance is too high, as shown by line 0b in Fig. 7.14, voltage

buildup will not be achieved. � The critical field resistance, corresponding to the slope of the field-resistance line

tangent to the magnetization curve, is the resistance above which buildup will not be obtained.

12

§7.6.2 Motor Analysis � The terminal voltage of a motor is usually held substantially constant or controlled to a specific

value. Motor analysis is most nearly resembles that for separately-excited generators. � Speed is an important variable and often the one whose value is to be found.

aaaa RIEV ±= (7.17) ( )saaat RRIEV +±= (7.18)

sf s

f

Gross mmf equivalent shunt-field amperesN

I IN

� �= + � �

� � (7.21)

adamech IKT Φ= (7.13)

mdaa ωΦ= KE (7.14)

ma0

m0aE E

ωω� �

= � �� �

(7.22)

00

aa Enn

E ���

����

�= (7.23)

13